<?xml version="1.0" encoding="UTF-8"?>
<QEDEQ
    xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
    xsi:noNamespaceSchemaLocation="http://www.qedeq.org/0_04_01/xml/qedeq.xsd">
  <HEADER email="mime@qedeq.org">
    <SPECIFICATION name="proof_002" ruleVersion="1.00.00">
      <LOCATIONS>
        <LOCATION value="http://www.qedeq.org/0_04_01/sample" />
      </LOCATIONS>
    </SPECIFICATION>
    <TITLE>
      <LATEX language="en">
        <![CDATA[
          First formal proofs.
        ]]>
      </LATEX>
    </TITLE>
    <ABSTRACT>
      <LATEX language="en">
        <![CDATA[
          Here are the proofs assembled. This module should be parseble although some parameters are missing.
        ]]>
      </LATEX>
    </ABSTRACT>
    <AUTHORS>
      <AUTHOR email="michael@meyling.com">
        <NAME>
          <LATEX language="en">
            <![CDATA[
              Michael Meyling
            ]]>
          </LATEX>
        </NAME>
      </AUTHOR>
    </AUTHORS>
  </HEADER>
  <CHAPTER>
    <TITLE>
      <LATEX language="en">
        <![CDATA[
          First
        ]]>
      </LATEX>
    </TITLE>
    <INTRODUCTION>
      <LATEX language="en">
        <![CDATA[
          After presenting some axioms we show some formal proofs.
        ]]>
      </LATEX>
    </INTRODUCTION>
    <SECTION>
      <TITLE>
        <LATEX language="en">
          <![CDATA[
            Axioms
          ]]>
        </LATEX>
      </TITLE>
      <INTRODUCTION>
        <LATEX language="en">
          <![CDATA[
            Here we state our axioms.
          ]]>
        </LATEX>
      </INTRODUCTION>
      <SUBSECTIONS>
        <NODE id="axiom:disjunction_idempotence">
          <NAME>
            <LATEX language="en">
              <![CDATA[
                axiom of disjunction idempotence
              ]]>
            </LATEX>
          </NAME>
          <TITLE>
            <LATEX language="en">
              <![CDATA[
                Disjunction Idempotence
              ]]>
            </LATEX>
          </TITLE>
          <AXIOM>
            <FORMULA>
              <IMPL>
                <OR>
                  <PREDVAR id="A"/>
                  <PREDVAR id="A"/>
                </OR>
                <PREDVAR id="A"/>
              </IMPL>
            </FORMULA>
          </AXIOM>
        </NODE>
        <NODE id="axiom:disjunction_weakening">
          <NAME>
            <LATEX language="en">
              <![CDATA[
                axiom of weakening
              ]]>
            </LATEX>
          </NAME>
          <TITLE>
            <LATEX language="en">
              <![CDATA[
                Axiom of Weakening
              ]]>
            </LATEX>
          </TITLE>
          <AXIOM>
            <FORMULA>
              <IMPL>
                <PREDVAR id="A"/>
                <OR>
                  <PREDVAR id="A"/>
                  <PREDVAR id="B"/>
                </OR>
              </IMPL>
            </FORMULA>
          </AXIOM>
        </NODE>
        <NODE id="axiom:disjunction_commutative">
          <NAME>
            <LATEX language="en">
              <![CDATA[
                commutativity of the disjunction
              ]]>
            </LATEX>
          </NAME>
          <TITLE>
            <LATEX language="en">
              <![CDATA[
                Commutativity of the Disjunction
              ]]>
            </LATEX>
          </TITLE>
          <AXIOM>
            <FORMULA>
              <IMPL>
                <OR>
                  <PREDVAR id="A"/>
                  <PREDVAR id="B"/>
                </OR>
                <OR>
                  <PREDVAR id="B"/>
                  <PREDVAR id="A"/>
                </OR>
              </IMPL>
            </FORMULA>
          </AXIOM>
        </NODE>
        <NODE id="axiom:disjunction_addition">
          <NAME>
            <LATEX language="en">
              <![CDATA[
                axiom of disjunctive addition
              ]]>
            </LATEX>
          </NAME>
          <TITLE>
            <LATEX language="en">
              <![CDATA[
                Disjunctive Addition
              ]]>
            </LATEX>
          </TITLE>
          <AXIOM>
            <FORMULA>
              <IMPL>
                <IMPL>
                  <PREDVAR id="A"/>
                  <PREDVAR id="B"/>
                </IMPL>
                <IMPL>
                  <IMPL>
                    <PREDVAR id="C"/>
                    <PREDVAR id="A"/>
                  </IMPL>
                  <IMPL>
                    <PREDVAR id="C"/>
                    <PREDVAR id="B"/>
                  </IMPL>
                </IMPL>
              </IMPL>
            </FORMULA>
          </AXIOM>
        </NODE>
        <NODE id="axiom:universalInstantiation">
          <NAME>
            <LATEX language="en">
              <![CDATA[
                axiom of universal instantiation
              ]]>
            </LATEX>
          </NAME>
          <TITLE>
            <LATEX language="en">
              <![CDATA[
                Universal Instantiation
              ]]>
            </LATEX>
          </TITLE>
          <AXIOM>
            <FORMULA>
              <IMPL>
                <FORALL>
                  <VAR id="x"/>
                  <PREDVAR id="\phi">
                    <VAR id="x"/>
                  </PREDVAR>
                </FORALL>
                <PREDVAR id="\phi">
                  <VAR id="y"/>
                </PREDVAR>
              </IMPL>
            </FORMULA>
          </AXIOM>
        </NODE>
        <NODE id="axiom:existencialGeneralization">
          <NAME>
            <LATEX language="en">
              <![CDATA[
                axiom of existential generalization
              ]]>
            </LATEX>
          </NAME>
          <TITLE>
            <LATEX language="en">
              <![CDATA[
                Existential Generalization
              ]]>
            </LATEX>
          </TITLE>
          <AXIOM>
            <FORMULA>
              <IMPL>
                <PREDVAR id="\phi">
                  <VAR id="y" />
                </PREDVAR>
                <EXISTS>
                  <VAR id="x" />
                  <PREDVAR id="\phi">
                    <VAR id="x" />
                  </PREDVAR>
                </EXISTS>
              </IMPL>
            </FORMULA>
          </AXIOM>
        </NODE>
      </SUBSECTIONS>
    </SECTION>
    <SECTION>
      <TITLE>
        <LATEX language="en">
          <![CDATA[
            One
          ]]>
        </LATEX>
      </TITLE>
      <INTRODUCTION>
        <LATEX language="en">
          <![CDATA[
            Here we start.
          ]]>
        </LATEX>
      </INTRODUCTION>
      <SUBSECTIONS>
        <NODE id="proposition:one" level="formal">
          <THEOREM>
            <FORMULA>
              <IMPL>
                <PREDVAR id="A"/>
                <PREDVAR id="A"/>
              </IMPL>
            </FORMULA>
            <FORMAL_PROOF>
              <LINES>
                <L label="1">
                  <FORMULA>
                    <IMPL>
                      <PREDVAR id="A"/>
                      <OR>
                        <PREDVAR id="A"/>
                        <PREDVAR id="B"/>
                      </OR>
                    </IMPL>
                  </FORMULA>
                  <!--  missing attribute "ref" -->
                  <ADD/>
                </L>
                <L label="2">
                  <FORMULA>
                    <IMPL>
                      <PREDVAR id="A"/>
                      <OR>
                        <PREDVAR id="A"/>
                        <PREDVAR id="A"/>
                      </OR>
                    </IMPL>
                  </FORMULA>
                  <!--  missing element "FORMULA" -->
                  <SUBST_PREDVAR ref="1">
                    <PREDVAR id="B"/>
                  </SUBST_PREDVAR>
                </L>
                <L label="3">
                  <FORMULA>
                    <IMPL>
                      <OR>
                        <PREDVAR id="A"/>
                        <PREDVAR id="A"/>
                      </OR>
                      <PREDVAR id="A"/>
                    </IMPL>
                  </FORMULA>
                  <!--  import reference don't exist -->
                  <ADD ref="dontExist.axiom:disjunction_idempotence"/>
                </L>
                <L label="4">
                  <FORMULA>
                    <IMPL>
                      <IMPL>
                        <PREDVAR id="A"/>
                        <PREDVAR id="B"/>
                      </IMPL>
                      <IMPL>
                        <IMPL>
                          <PREDVAR id="C"/>
                          <PREDVAR id="A"/>
                        </IMPL>
                        <IMPL>
                          <PREDVAR id="C"/>
                          <PREDVAR id="B"/>
                        </IMPL>
                      </IMPL>
                    </IMPL>
                  </FORMULA>
                  <!--  missing node reference -->
                  <ADD ref="axion:disjunction_addition"/>
                </L>
                <L label="5">
                  <FORMULA>
                    <IMPL>
                      <IMPL>
                        <PREDVAR id="D"/>
                        <PREDVAR id="B"/>
                      </IMPL>
                      <IMPL>
                        <IMPL>
                          <PREDVAR id="C"/>
                          <PREDVAR id="D"/>
                        </IMPL>
                        <IMPL>
                          <PREDVAR id="C"/>
                          <PREDVAR id="B"/>
                        </IMPL>
                      </IMPL>
                    </IMPL>
                  </FORMULA>
                  <!--  missing element "PREDVAR" -->
                  <SUBST_PREDVAR ref="4">
                    <FORMULA>
                      <PREDVAR id="D"/>
                    </FORMULA>
                  </SUBST_PREDVAR>
                </L>
                <L label="6">
                  <FORMULA>
                    <IMPL>
                      <IMPL>
                        <PREDVAR id="D"/>
                        <PREDVAR id="A"/>
                      </IMPL>
                      <IMPL>
                        <IMPL>
                          <PREDVAR id="C"/>
                          <PREDVAR id="D"/>
                        </IMPL>
                        <IMPL>
                          <PREDVAR id="C"/>
                          <PREDVAR id="A"/>
                        </IMPL>
                      </IMPL>
                    </IMPL>
                  </FORMULA>
                  <!--  missing element "FORMULA" and "PREDVAR" -->
                  <SUBST_PREDVAR ref="5">
                  </SUBST_PREDVAR>
                </L>
                <L label="7">
                  <FORMULA>
                    <IMPL>
                      <IMPL>
                        <PREDVAR id="D"/>
                        <PREDVAR id="A"/>
                      </IMPL>
                      <IMPL>
                        <IMPL>
                          <PREDVAR id="A"/>
                          <PREDVAR id="D"/>
                        </IMPL>
                        <IMPL>
                          <PREDVAR id="A"/>
                          <PREDVAR id="A"/>
                        </IMPL>
                      </IMPL>
                    </IMPL>
                  </FORMULA>
                  <!--  missing attribute "ref" -->
                  <SUBST_PREDVAR>
                    <PREDVAR id="C"/>
                    <FORMULA>
                      <PREDVAR id="A"/>
                    </FORMULA>
                  </SUBST_PREDVAR>
                </L>
                <L label="8">
                  <FORMULA>
                    <IMPL>
                      <IMPL>
                        <OR>
                          <PREDVAR id="A"/>
                          <PREDVAR id="A"/>
                        </OR>
                        <PREDVAR id="A"/>
                      </IMPL>
                      <IMPL>
                        <IMPL>
                          <PREDVAR id="A"/>
                          <OR>
                            <PREDVAR id="A"/>
                            <PREDVAR id="A"/>
                          </OR>
                        </IMPL>
                        <IMPL>
                          <PREDVAR id="A"/>
                          <PREDVAR id="A"/>
                        </IMPL>
                      </IMPL>
                    </IMPL>
                  </FORMULA>
                  <!--  missing attribute "ref" -->
                  <SUBST_PREDVAR>
                    <PREDVAR id="D"/>
                    <FORMULA>
                      <OR>
                        <PREDVAR id="A"/>
                        <PREDVAR id="A"/>
                      </OR>
                    </FORMULA>
                  </SUBST_PREDVAR>
                </L>
                <!--  missing attribute "label" -->
                <L>
                  <FORMULA>
                    <IMPL>
                      <IMPL>
                        <PREDVAR id="A"/>
                        <OR>
                          <PREDVAR id="A"/>
                          <PREDVAR id="A"/>
                        </OR>
                      </IMPL>
                      <IMPL>
                        <PREDVAR id="A"/>
                        <PREDVAR id="A"/>
                      </IMPL>
                    </IMPL>
                  </FORMULA>
                  <!--  missing attribute "ref1" -->
                  <MP ref2="3"/>
                </L>
                <L label="10">
                  <FORMULA>
                    <IMPL>
                      <PREDVAR id="A"/>
                      <PREDVAR id="A"/>
                    </IMPL>
                  </FORMULA>
                  <!--  missing attribute "ref2" -->
                  <MP ref1="9"/>
                </L>
              </LINES>
            </FORMAL_PROOF>
          </THEOREM>
        </NODE>
        <NODE id="proposition:two" level="formal">
          <THEOREM>
            <FORMULA>
              <IMPL>
                <FORALL>
                  <VAR id="x"/>
                  <PREDVAR id="\phi">
                    <VAR id="x" />
                  </PREDVAR>
                </FORALL>
                <EXISTS>
                  <VAR id="x" />
                  <PREDVAR id="\phi">
                    <VAR id="x" />
                  </PREDVAR>
                </EXISTS>
              </IMPL>
            </FORMULA>
            <FORMAL_PROOF>
              <LINES>
                <L label="1">
                  <FORMULA>
                    <IMPL>
                      <FORALL>
                        <VAR id="x"/>
                        <PREDVAR id="\phi">
                          <VAR id="x" />
                        </PREDVAR>
                      </FORALL>
                      <PREDVAR id="\phi">
                        <VAR id="y" />
                      </PREDVAR>
                    </IMPL>
                  </FORMULA>
                  <ADD ref="axiom:universalInstantiation"/>
                </L>
                <L label="2">
                  <FORMULA>
                    <IMPL>
                      <PREDVAR id="\phi">
                        <VAR id="y" />
                      </PREDVAR>
                      <EXISTS>
                        <VAR id="x" />
                        <PREDVAR id="\phi">
                          <VAR id="x" />
                        </PREDVAR>
                      </EXISTS>
                    </IMPL>
                    </FORMULA>
                  <!--  attribute "ref" has incorrect reference -->
                  <ADD ref="axiom:existencialGeneralization!1"/>
                </L>
                <L label="3">
                  <FORMULA>
                    <IMPL>
                      <IMPL>
                        <PREDVAR id="A"/>
                        <PREDVAR id="B"/>
                      </IMPL>
                      <IMPL>
                        <IMPL>
                          <PREDVAR id="C"/>
                          <PREDVAR id="A"/>
                        </IMPL>
                        <IMPL>
                          <PREDVAR id="C"/>
                          <PREDVAR id="B"/>
                        </IMPL>
                      </IMPL>
                    </IMPL>
                  </FORMULA>
                  <!--  attribute "ref" has incorrect reference -->
                  <ADD ref="axiom:disjunction_addition/a"/>
                </L>
                <L label="4">
                  <FORMULA>
                    <IMPL>
                      <IMPL>
                        <PREDVAR id="\phi">
                          <VAR id="y" />
                        </PREDVAR>
                        <PREDVAR id="B"/>
                      </IMPL>
                      <IMPL>
                        <IMPL>
                          <PREDVAR id="C"/>
                        <PREDVAR id="\phi">
                          <VAR id="y" />
                        </PREDVAR>
                        </IMPL>
                        <IMPL>
                          <PREDVAR id="C"/>
                          <PREDVAR id="B"/>
                        </IMPL>
                      </IMPL>
                    </IMPL>
                  </FORMULA>
                  <!--  no attributes or references -->
                  <SUBST_PREDVAR />
                </L>
                <L label="5">
                  <FORMULA>
                    <IMPL>
                      <IMPL>
                        <PREDVAR id="\phi">
                          <VAR id="y" />
                        </PREDVAR>
                        <PREDVAR id="B"/>
                      </IMPL>
                      <IMPL>
                        <IMPL>
                          <FORALL>
                            <VAR id="x" />
                            <PREDVAR id="\phi">
                              <VAR id="x" />
                            </PREDVAR>
                          </FORALL>
                          <PREDVAR id="\phi">
                            <VAR id="y" />
                          </PREDVAR>
                        </IMPL>
                        <IMPL>
                          <FORALL>
                            <VAR id="x" />
                            <PREDVAR id="\phi">
                              <VAR id="x" />
                            </PREDVAR>
                          </FORALL>
                          <PREDVAR id="B"/>
                        </IMPL>
                      </IMPL>
                    </IMPL>
                  </FORMULA>
                  <SUBST_PREDVAR ref="4">
                    <PREDVAR id="C"/>
                    <FORMULA>
                      <FORALL>
                        <VAR id="x" />
                        <PREDVAR id="\phi">
                          <VAR id="x" />
                        </PREDVAR>
                      </FORALL>
                    </FORMULA>
                  </SUBST_PREDVAR>
                </L>
                <L label="6">
                  <FORMULA>
                    <IMPL>
                      <IMPL>
                        <PREDVAR id="\phi">
                          <VAR id="y" />
                        </PREDVAR>
                        <EXISTS>
                          <VAR id="x" />
                          <PREDVAR id="\phi">
                            <VAR id="x" />
                          </PREDVAR>
                        </EXISTS>
                      </IMPL>
                      <IMPL>
                        <IMPL>
                          <FORALL>
                            <VAR id="x" />
                            <PREDVAR id="\phi">
                              <VAR id="x" />
                            </PREDVAR>
                          </FORALL>
                          <PREDVAR id="\phi">
                            <VAR id="y" />
                          </PREDVAR>
                        </IMPL>
                        <IMPL>
                          <FORALL>
                            <VAR id="x" />
                            <PREDVAR id="\phi">
                              <VAR id="x" />
                            </PREDVAR>
                          </FORALL>
                          <EXISTS>
                            <VAR id="x" />
                            <PREDVAR id="\phi">
                              <VAR id="x" />
                            </PREDVAR>
                          </EXISTS>
                        </IMPL>
                      </IMPL>
                    </IMPL>
                  </FORMULA>
                  <SUBST_PREDVAR ref="5">
                    <PREDVAR id="B"/>
                    <FORMULA>
                      <EXISTS>
                        <VAR id="x" />
                        <PREDVAR id="\phi">
                          <VAR id="x" />
                        </PREDVAR>
                      </EXISTS>
                    </FORMULA>
                  </SUBST_PREDVAR>
                </L>
                <L label="7">
                  <FORMULA>
                    <IMPL>
                      <IMPL>
                        <FORALL>
                          <VAR id="x" />
                          <PREDVAR id="\phi">
                            <VAR id="x" />
                          </PREDVAR>
                        </FORALL>
                        <PREDVAR id="\phi">
                          <VAR id="y" />
                        </PREDVAR>
                      </IMPL>
                      <IMPL>
                        <FORALL>
                          <VAR id="x" />
                          <PREDVAR id="\phi">
                            <VAR id="x" />
                          </PREDVAR>
                        </FORALL>
                        <EXISTS>
                          <VAR id="x" />
                          <PREDVAR id="\phi">
                            <VAR id="x" />
                          </PREDVAR>
                        </EXISTS>
                      </IMPL>
                    </IMPL>
                  </FORMULA>
                  <!--  attributes "ref1" and "ref2" are missing -->
                  <MP />
                </L>
                <L label="8">
                  <FORMULA>
                    <IMPL>
                      <FORALL>
                        <VAR id="x" />
                        <PREDVAR id="\phi">
                          <VAR id="x" />
                        </PREDVAR>
                      </FORALL>
                      <EXISTS>
                        <VAR id="x" />
                        <PREDVAR id="\phi">
                          <VAR id="x" />
                        </PREDVAR>
                      </EXISTS>
                    </IMPL>
                  </FORMULA>
                  <MP ref1="7" ref2="1" />
                </L>
              </LINES>
            </FORMAL_PROOF>
          </THEOREM>
        </NODE>
        <NODE id="proposition:three" level="formal">
          <THEOREM>
            <FORMULA>
              <IMPL>
                <EXISTS>
                  <VAR id="x" />
                  <FORALL>
                    <VAR id="y"/>
                    <PREDVAR id="\phi">
                      <VAR id="x" />
                      <VAR id="y" />
                    </PREDVAR>
                  </FORALL>
                </EXISTS>
                <FORALL>
                  <VAR id="y"/>
                  <EXISTS>
                    <VAR id="x" />
                    <PREDVAR id="\phi">
                      <VAR id="x" />
                      <VAR id="y" />
                    </PREDVAR>
                  </EXISTS>
                </FORALL>
              </IMPL>
            </FORMULA>
            <FORMAL_PROOF>
              <LINES>
                <L label="1">
                  <FORMULA>
                    <IMPL>
                      <FORALL>
                        <VAR id="x" />
                        <PREDVAR id="\phi">
                          <VAR id="x" />
                        </PREDVAR>
                      </FORALL>
                      <EXISTS>
                        <VAR id="x" />
                        <PREDVAR id="\phi">
                          <VAR id="x" />
                        </PREDVAR>
                      </EXISTS>
                    </IMPL>
                  </FORMULA>
                  <!--  attributes "ref" is empty -->
                  <ADD ref="" />
                </L>
                <L label="2">
                  <FORMULA>
                    <IMPL>
                      <FORALL>
                        <VAR id="x" />
                        <PREDVAR id="\phi">
                          <VAR id="x" />
                        </PREDVAR>
                      </FORALL>
                      <EXISTS>
                        <VAR id="y" />
                        <PREDVAR id="\phi">
                          <VAR id="y" />
                        </PREDVAR>
                      </EXISTS>
                    </IMPL>
                  </FORMULA>
                  <!--  attributes "ref" and "occurrence" are missing -->
                  <RENAME>
                    <VAR id="x" />
                    <VAR id="y" />
                  </RENAME>
                </L>
                <L label="3">
                  <FORMULA>
                    <IMPL>
                      <FORALL>
                        <VAR id="y" />
                        <PREDVAR id="\phi">
                          <VAR id="y" />
                        </PREDVAR>
                      </FORALL>
                      <EXISTS>
                        <VAR id="y" />
                        <PREDVAR id="\phi">
                          <VAR id="y" />
                        </PREDVAR>
                      </EXISTS>
                    </IMPL>
                  </FORMULA>
                  <!--  VAR elements are missing -->
                  <RENAME ref="2" occurrence="1">
                  </RENAME>
                </L>
                <L label="4">
                  <FORMULA>
                    <IMPL>
                      <FORALL>
                        <VAR id="y" />
                        <PREDVAR id="\phi">
                          <VAR id="x" />
                          <VAR id="y" />
                        </PREDVAR>
                      </FORALL>
                      <EXISTS>
                        <VAR id="y" />
                        <PREDVAR id="\phi">
                          <VAR id="x" />
                          <VAR id="y" />
                        </PREDVAR>
                      </EXISTS>
                    </IMPL>
                  </FORMULA>
                  <!--  attributes and elements are missing -->
                  <SUBST_PREDVAR />
                </L>
                <L label="5">
                  <FORMULA>
                    <IMPL>
                      <EXISTS>
                        <VAR id="x" />
                          <FORALL>
                            <VAR id="y" />
                            <PREDVAR id="\phi">
                              <VAR id="x" />
                              <VAR id="y" />
                            </PREDVAR>
                          </FORALL>
                      </EXISTS>
                      <EXISTS>
                        <VAR id="y" />
                        <PREDVAR id="\phi">
                          <VAR id="x" />
                          <VAR id="y" />
                        </PREDVAR>
                      </EXISTS>
                    </IMPL>
                  </FORMULA>
                  <!--  attributes and elements are missing -->
                  <EXISTENTIAL />
                </L>
                <L label="6">
                  <FORMULA>
                    <IMPL>
                      <EXISTS>
                        <VAR id="x" />
                          <FORALL>
                            <VAR id="y" />
                            <PREDVAR id="\phi">
                              <VAR id="x" />
                              <VAR id="y" />
                            </PREDVAR>
                          </FORALL>
                      </EXISTS>
                      <FORALL>
                        <VAR id="x" />
                        <EXISTS>
                          <VAR id="y" />
                          <PREDVAR id="\phi">
                            <VAR id="x" />
                            <VAR id="y" />
                          </PREDVAR>
                        </EXISTS>
                      </FORALL>
                    </IMPL>
                  </FORMULA>
                  <!--  attributes and elements are missing -->
                  <UNIVERSAL />
                </L>
              </LINES>
            </FORMAL_PROOF>
         </THEOREM>
        </NODE>
        <NODE id="proposition:four" level="formal">
          <THEOREM>
            <FORMULA>
              <IMPL>
                <FORALL>
                  <VAR id="x" />
                  <FORALL>
                    <VAR id="y"/>
                    <PREDVAR id="\phi">
                      <VAR id="x" />
                      <VAR id="y" />
                    </PREDVAR>
                  </FORALL>
                </FORALL>
                <FORALL>
                  <VAR id="y"/>
                  <FORALL>
                    <VAR id="x" />
                    <PREDVAR id="\phi">
                      <VAR id="x" />
                      <VAR id="y" />
                    </PREDVAR>
                  </FORALL>
                </FORALL>
              </IMPL>
            </FORMULA>
            <FORMAL_PROOF>
              <LINES>
                <L label="1">
                  <FORMULA>
                    <IMPL>
                      <FORALL>
                        <VAR id="x"/>
                        <PREDVAR id="\phi">
                          <VAR id="x"/>
                        </PREDVAR>
                      </FORALL>
                      <PREDVAR id="\phi">
                        <VAR id="y"/>
                      </PREDVAR>
                    </IMPL>
                  </FORMULA>
                  <ADD ref="axiom:universalInstantiation" />
                </L>
                <L label="2">
                  <FORMULA>
                    <IMPL>
                      <FORALL>
                        <VAR id="x"/>
                        <PREDVAR id="\phi">
                          <VAR id="x"/>
                        </PREDVAR>
                      </FORALL>
                      <PREDVAR id="\phi">
                        <VAR id="u"/>
                      </PREDVAR>
                    </IMPL>
                  </FORMULA>
                  <!--  attributes and elements are missing -->
                  <SUBST_FREE />
                </L>
                <L label="3">
                  <FORMULA>
                    <IMPL>
                      <FORALL>
                        <VAR id="y"/>
                        <PREDVAR id="\phi">
                          <VAR id="y"/>
                        </PREDVAR>
                      </FORALL>
                      <PREDVAR id="\phi">
                        <VAR id="u"/>
                      </PREDVAR>
                    </IMPL>
                  </FORMULA>
                  <!--  attribute "occurrence" and one "VAR" element are missing -->
                  <RENAME ref="2">
                    <VAR id="x" />
                  </RENAME>
                </L>
                <L label="4">
                  <FORMULA>
                    <IMPL>
                      <FORALL>
                        <VAR id="y"/>
                        <PREDVAR id="\phi">
                          <VAR id="z"/>
                          <VAR id="y"/>
                        </PREDVAR>
                      </FORALL>
                      <PREDVAR id="\phi">
                        <VAR id="z"/>
                        <VAR id="u"/>
                      </PREDVAR>
                    </IMPL>
                  </FORMULA>
                  <!--  attribute "ref" and "PREDVAR" element are missing -->
                  <SUBST_PREDVAR>
                    <FORMULA>
                      <PREDVAR id="\phi">
                        <VAR id="z" />
                        <VAR id="x" />
                      </PREDVAR>
                    </FORMULA>
                  </SUBST_PREDVAR>
                </L>
                <L label="5">
                  <FORMULA>
                    <IMPL>
                      <FORALL>
                        <VAR id="x"/>
                        <FORALL>
                          <VAR id="y"/>
                          <PREDVAR id="\phi">
                            <VAR id="x"/>
                            <VAR id="y"/>
                          </PREDVAR>
                        </FORALL>
                      </FORALL>
                      <FORALL>
                        <VAR id="y"/>
                        <PREDVAR id="\phi">
                          <VAR id="u"/>
                          <VAR id="y"/>
                        </PREDVAR>
                      </FORALL>
                    </IMPL>
                  </FORMULA>
                  <!--  "ref" attribute and "FORMULA" element are missing -->
                  <SUBST_PREDVAR>
                    <PREDVAR id="\phi">
                      <VAR id="x" />
                    </PREDVAR>
                  </SUBST_PREDVAR>
                </L>
                <L label="6">
                  <FORMULA>
                    <IMPL>
                      <FORALL>
                        <VAR id="x"/>
                        <FORALL>
                          <VAR id="y"/>
                          <PREDVAR id="\phi">
                            <VAR id="x"/>
                            <VAR id="y"/>
                          </PREDVAR>
                        </FORALL>
                      </FORALL>
                      <FORALL>
                        <VAR id="y"/>
                        <PREDVAR id="\phi">
                          <VAR id="z"/>
                          <VAR id="y"/>
                        </PREDVAR>
                      </FORALL>
                    </IMPL>
                  </FORMULA>
                  <!--  element "VAR" element is missing -->
                  <SUBST_FREE ref="5">
                    <TERM>
                      <VAR id="z" />
                    </TERM>
                  </SUBST_FREE>
                </L>
                <L label="7">
                  <FORMULA>
                    <IMPL>
                      <IMPL>
                        <PREDVAR id="A"/>
                        <PREDVAR id="B"/>
                      </IMPL>
                      <IMPL>
                        <IMPL>
                          <PREDVAR id="C"/>
                          <PREDVAR id="A"/>
                        </IMPL>
                        <IMPL>
                          <PREDVAR id="C"/>
                          <PREDVAR id="B"/>
                        </IMPL>
                      </IMPL>
                    </IMPL>
                  </FORMULA>
                  <ADD ref="axiom:disjunction_addition"/>
                </L>
                <L label="8">
                  <FORMULA>
                    <IMPL>
                      <IMPL>
                        <FORALL>
                          <VAR id="y" />
                          <PREDVAR id="\phi">
                            <VAR id="z" />
                            <VAR id="y" />
                          </PREDVAR>
                        </FORALL>
                        <PREDVAR id="B"/>
                      </IMPL>
                      <IMPL>
                        <IMPL>
                          <PREDVAR id="C"/>
                          <FORALL>
                            <VAR id="y" />
                            <PREDVAR id="\phi">
                              <VAR id="z" />
                              <VAR id="y" />
                            </PREDVAR>
                          </FORALL>
                        </IMPL>
                        <IMPL>
                          <PREDVAR id="C"/>
                          <PREDVAR id="B"/>
                        </IMPL>
                      </IMPL>
                    </IMPL>
                  </FORMULA>
                  <!--  element "PREDVAR" and "FORMULA" are missing -->
                  <SUBST_PREDVAR ref="7">
                  </SUBST_PREDVAR>
                </L>
                <L label="9">
                  <FORMULA>
                    <IMPL>
                      <IMPL>
                        <FORALL>
                          <VAR id="y" />
                          <PREDVAR id="\phi">
                            <VAR id="z" />
                            <VAR id="y" />
                          </PREDVAR>
                        </FORALL>
                        <PREDVAR id="\phi">
                          <VAR id="z" />
                          <VAR id="u" />
                        </PREDVAR>
                      </IMPL>
                      <IMPL>
                        <IMPL>
                          <PREDVAR id="C"/>
                          <FORALL>
                            <VAR id="y" />
                            <PREDVAR id="\phi">
                              <VAR id="z" />
                              <VAR id="y" />
                            </PREDVAR>
                          </FORALL>
                        </IMPL>
                        <IMPL>
                          <PREDVAR id="C"/>
                          <PREDVAR id="\phi">
                            <VAR id="z" />
                            <VAR id="u" />
                          </PREDVAR>
                        </IMPL>
                      </IMPL>
                    </IMPL>
                  </FORMULA>
                  <!--  attribute "ref" has wrong content -->
                  <SUBST_PREDVAR ref="8.8.8">
                    <PREDVAR id="B" />
                    <FORMULA>
                      <PREDVAR id="\phi">
                        <VAR id="z" />
                        <VAR id="u" />
                      </PREDVAR>
                    </FORMULA>
                  </SUBST_PREDVAR>
                </L>
                <L label="10">
                  <FORMULA>
                    <IMPL>
                      <IMPL>
                        <FORALL>
                          <VAR id="y" />
                          <PREDVAR id="\phi">
                            <VAR id="z" />
                            <VAR id="y" />
                          </PREDVAR>
                        </FORALL>
                        <PREDVAR id="\phi">
                          <VAR id="z" />
                          <VAR id="u" />
                        </PREDVAR>
                      </IMPL>
                      <IMPL>
                        <IMPL>
                          <FORALL>
                            <VAR id="x" />
                            <FORALL>
                              <VAR id="y" />
                              <PREDVAR id="\phi">
                                <VAR id="x" />
                                <VAR id="y" />
                              </PREDVAR>
                            </FORALL>
                          </FORALL>
                          <FORALL>
                            <VAR id="y" />
                            <PREDVAR id="\phi">
                              <VAR id="z" />
                              <VAR id="y" />
                            </PREDVAR>
                          </FORALL>
                        </IMPL>
                        <IMPL>
                          <FORALL>
                            <VAR id="x" />
                            <FORALL>
                              <VAR id="y" />
                              <PREDVAR id="\phi">
                                <VAR id="x" />
                                <VAR id="y" />
                              </PREDVAR>
                            </FORALL>
                          </FORALL>
                          <PREDVAR id="\phi">
                            <VAR id="z" />
                            <VAR id="u" />
                          </PREDVAR>
                        </IMPL>
                      </IMPL>
                    </IMPL>
                  </FORMULA>
                  <!--  no attributes and elements -->
                  <SUBST_PREDVAR />
                </L>
                <L label="11">
                  <FORMULA>
                    <IMPL>
                      <IMPL>
                        <FORALL>
                          <VAR id="x" />
                          <FORALL>
                            <VAR id="y" />
                            <PREDVAR id="\phi">
                              <VAR id="x" />
                              <VAR id="y" />
                            </PREDVAR>
                          </FORALL>
                        </FORALL>
                        <FORALL>
                          <VAR id="y" />
                          <PREDVAR id="\phi">
                            <VAR id="z" />
                            <VAR id="y" />
                          </PREDVAR>
                        </FORALL>
                      </IMPL>
                      <IMPL>
                        <FORALL>
                          <VAR id="x" />
                          <FORALL>
                            <VAR id="y" />
                            <PREDVAR id="\phi">
                              <VAR id="x" />
                              <VAR id="y" />
                            </PREDVAR>
                          </FORALL>
                        </FORALL>
                        <PREDVAR id="\phi">
                          <VAR id="z" />
                          <VAR id="u" />
                        </PREDVAR>
                      </IMPL>
                    </IMPL>
                  </FORMULA>
                  <!--  references are invalid -->
                  <MP ref1="10.1.2" ref2="4.4.1" />
                </L>
                <L label="12">
                  <FORMULA>
                    <IMPL>
                      <FORALL>
                        <VAR id="x" />
                        <FORALL>
                          <VAR id="y" />
                          <PREDVAR id="\phi">
                            <VAR id="x" />
                            <VAR id="y" />
                          </PREDVAR>
                        </FORALL>
                      </FORALL>
                      <PREDVAR id="\phi">
                        <VAR id="z" />
                        <VAR id="u" />
                      </PREDVAR>
                    </IMPL>
                  </FORMULA>
                  <MP ref1="11" ref2="6" />
                </L>
                <L label="13">
                  <FORMULA>
                    <IMPL>
                      <FORALL>
                        <VAR id="x" />
                        <FORALL>
                          <VAR id="y" />
                          <PREDVAR id="\phi">
                            <VAR id="x" />
                            <VAR id="y" />
                          </PREDVAR>
                        </FORALL>
                      </FORALL>
                      <FORALL>
                        <VAR id="z" />
                        <PREDVAR id="\phi">
                          <VAR id="z" />
                          <VAR id="u" />
                        </PREDVAR>
                      </FORALL>
                    </IMPL>
                  </FORMULA>
                  <!--  element "VAR" is missing -->
                  <UNIVERSAL ref="12">
                  </UNIVERSAL>
                </L>
                <L label="13">
                  <FORMULA>
                    <IMPL>
                      <FORALL>
                        <VAR id="x" />
                        <FORALL>
                          <VAR id="y" />
                          <PREDVAR id="\phi">
                            <VAR id="x" />
                            <VAR id="y" />
                          </PREDVAR>
                        </FORALL>
                      </FORALL>
                      <FORALL>
                        <VAR id="u" />
                        <FORALL>
                          <VAR id="z" />
                          <PREDVAR id="\phi">
                            <VAR id="z" />
                            <VAR id="u" />
                          </PREDVAR>
                        </FORALL>
                      </FORALL>
                    </IMPL>
                  </FORMULA>
                  <!--  element "VAR" is missing -->
                  <UNIVERSAL ref="12">
                  </UNIVERSAL>
                </L>
                <L label="13">
                  <FORMULA>
                    <IMPL>
                      <FORALL>
                        <VAR id="x" />
                        <FORALL>
                          <VAR id="y" />
                          <PREDVAR id="\phi">
                            <VAR id="x" />
                            <VAR id="y" />
                          </PREDVAR>
                        </FORALL>
                      </FORALL>
                      <FORALL>
                        <VAR id="y" />
                        <FORALL>
                          <VAR id="z" />
                          <PREDVAR id="\phi">
                            <VAR id="z" />
                            <VAR id="y" />
                          </PREDVAR>
                        </FORALL>
                      </FORALL>
                    </IMPL>
                  </FORMULA>
                  <!--  all "VAR" elements are missing -->
                  <RENAME ref="12" occurrence="1">
                  </RENAME>
                </L>
                <L label="14">
                  <FORMULA>
                    <IMPL>
                      <FORALL>
                        <VAR id="x" />
                        <FORALL>
                          <VAR id="y" />
                          <PREDVAR id="\phi">
                            <VAR id="x" />
                            <VAR id="y" />
                          </PREDVAR>
                        </FORALL>
                      </FORALL>
                      <FORALL>
                        <VAR id="y" />
                        <FORALL>
                          <VAR id="x" />
                          <PREDVAR id="\phi">
                            <VAR id="x" />
                            <VAR id="y" />
                          </PREDVAR>
                        </FORALL>
                      </FORALL>
                    </IMPL>
                  </FORMULA>
                  <!--  "ref" and "occurrence" attributes are missing -->
                  <RENAME>
                    <VAR id="z" />
                    <VAR id="x" />
                  </RENAME>
                </L>
              </LINES>
            </FORMAL_PROOF>
          </THEOREM>
        </NODE>
        <NODE id="proposition:five" level="formal">
          <THEOREM>
            <FORMULA>
              <IMPL>
                <FORALL>
                  <VAR id="x"/>
                  <PREDVAR id="\phi">
                    <VAR id="x" />
                  </PREDVAR>
                </FORALL>
                <FORALL>
                  <VAR id="x" />
                  <PREDVAR id="\phi">
                    <FUNVAR id="f">
                      <VAR id="x" />
                    </FUNVAR>
                  </PREDVAR>
                </FORALL>
              </IMPL>
            </FORMULA>
            <FORMAL_PROOF>
              <LINES>
                <L label="1">
                  <FORMULA>
                    <IMPL>
                      <FORALL>
                        <VAR id="x"/>
                        <PREDVAR id="\phi">
                          <VAR id="x" />
                        </PREDVAR>
                      </FORALL>
                      <PREDVAR id="\phi">
                        <VAR id="y" />
                      </PREDVAR>
                    </IMPL>
                  </FORMULA>
                  <ADD ref="axiom:universalInstantiation"/>
                </L>
                <L label="2">
                  <FORMULA>
                    <IMPL>
                      <FORALL>
                        <VAR id="x"/>
                        <PREDVAR id="\phi">
                          <VAR id="x" />
                        </PREDVAR>
                      </FORALL>
                      <PREDVAR id="\phi">
                        <FUNVAR id="g">
                          <VAR id="y" />
                        </FUNVAR>
                      </PREDVAR>
                    </IMPL>
                  </FORMULA>
                  <!--  "TERM" element is missing -->
                  <SUBST_FREE ref="1">
                     <TERM>
                       <FUNVAR id="g">
                         <VAR id="y" />
                       </FUNVAR>
                     </TERM>
                  </SUBST_FREE>
                </L>
                <L label="3">
                  <FORMULA>
                    <IMPL>
                      <FORALL>
                        <VAR id="x"/>
                        <PREDVAR id="\phi">
                          <VAR id="x" />
                        </PREDVAR>
                      </FORALL>
                      <PREDVAR id="\phi">
                        <FUNVAR id="f">
                          <VAR id="y" />
                        </FUNVAR>
                      </PREDVAR>
                    </IMPL>
                  </FORMULA>
                  <!--  all attributes and elements are missing -->
                  <SUBST_FUNVAR />
                </L>
                <L label="4">
                  <FORMULA>
                    <IMPL>
                      <FORALL>
                        <VAR id="x"/>
                        <PREDVAR id="\phi">
                          <VAR id="x" />
                        </PREDVAR>
                      </FORALL>
                      <FORALL>
                        <VAR id="y" />
                        <PREDVAR id="\phi">
                          <FUNVAR id="f">
                            <VAR id="y" />
                          </FUNVAR>
                        </PREDVAR>
                      </FORALL>
                    </IMPL>
                  </FORMULA>
                  <!--  "ref" attribute is missing -->
                  <UNIVERSAL>
                    <VAR id="y" />
                  </UNIVERSAL>
                </L>
                <L label="4">
                  <FORMULA>
                    <IMPL>
                      <FORALL>
                        <VAR id="x"/>
                        <PREDVAR id="\phi">
                          <VAR id="x" />
                        </PREDVAR>
                      </FORALL>
                      <FORALL>
                        <VAR id="x" />
                        <PREDVAR id="\phi">
                          <FUNVAR id="f">
                            <VAR id="x" />
                          </FUNVAR>
                        </PREDVAR>
                      </FORALL>
                    </IMPL>
                  </FORMULA>
                  <RENAME ref="4">
                    <VAR id="y" />
                    <VAR id="x" />
                  </RENAME>
                </L>
              </LINES>
            </FORMAL_PROOF>
          </THEOREM>
        </NODE>
        <NODE id="proposition:six" level="formal">
          <PRECEDING>
            <LATEX language="en">
              No useful proof, but uses missing rules.
            </LATEX>
          </PRECEDING>
          <THEOREM>
            <FORMULA>
              <IMPL>
                <FORALL>
                  <VAR id="x"/>
                  <PREDVAR id="\phi">
                    <VAR id="x" />
                  </PREDVAR>
                </FORALL>
                <EXISTS>
                  <VAR id="x" />
                  <PREDVAR id="\phi">
                    <VAR id="x" />
                  </PREDVAR>
                </EXISTS>
              </IMPL>
            </FORMULA>
            <FORMAL_PROOF>
              <LINES>
                <L label="1">
                  <FORMULA>
                    <IMPL>
                      <FORALL>
                        <VAR id="x" />
                        <PREDVAR id="\phi">
                          <VAR id="x" />
                        </PREDVAR>
                      </FORALL>
                      <EXISTS>
                        <VAR id="x" />
                        <PREDVAR id="\phi">
                          <VAR id="x" />
                        </PREDVAR>
                      </EXISTS>
                    </IMPL>
                  </FORMULA>
                  <ADD ref="proposition:two" />
                </L>
                <L label="2">
                  <FORMULA>
                    <IMPL>
                      <FORALL>
                        <VAR id="x" />
                        <PREDVAR id="\phi">
                          <VAR id="x" />
                        </PREDVAR>
                      </FORALL>
                      <EXISTS>
                        <VAR id="y" />
                        <PREDVAR id="\phi">
                          <VAR id="y" />
                        </PREDVAR>
                      </EXISTS>
                    </IMPL>
                  </FORMULA>
                  <RENAME ref="1">
                    <VAR id="x" />
                    <VAR id="y" />
                  </RENAME>
                </L>
                <L label="3">
                  <FORMULA>
                    <IMPL>
                      <FORALL>
                        <VAR id="x" />
                        <PREDVAR id="\phi">
                          <VAR id="z" />
                          <VAR id="x" />
                        </PREDVAR>
                      </FORALL>
                      <EXISTS>
                        <VAR id="y" />
                        <PREDVAR id="\phi">
                          <VAR id="z" />
                          <VAR id="y" />
                        </PREDVAR>
                      </EXISTS>
                    </IMPL>
                  </FORMULA>
                  <SUBST_PREDVAR ref="2">
                      <PREDVAR id="\phi">
                        <VAR id="u" />
                      </PREDVAR>
                    <FORMULA>
                      <PREDVAR id="\phi">
                        <VAR id="z" />
                        <VAR id="u" />
                      </PREDVAR>
                    </FORMULA>
                  </SUBST_PREDVAR>
                </L>
                <L label="4">
                  <FORMULA>
                    <IMPL>
                      <FORALL>
                        <VAR id="x" />
                        <PREDVAR id="\phi">
                          <FUNVAR id="f">
                            <VAR id="z" />
                          </FUNVAR>
                          <VAR id="x" />
                        </PREDVAR>
                      </FORALL>
                      <EXISTS>
                        <VAR id="y" />
                        <PREDVAR id="\phi">
                          <FUNVAR id="f">
                            <VAR id="z" />
                          </FUNVAR>
                          <VAR id="y" />
                        </PREDVAR>
                      </EXISTS>
                    </IMPL>
                  </FORMULA>
                  <SUBST_FREE ref="3">
                    <VAR id="u" />
                    <TERM>
                      <FUNVAR id="f">
                        <VAR id="u" />
                      </FUNVAR>
                    </TERM>
                  </SUBST_FREE>
                </L>
                <L label="5">
                  <FORMULA>
                    <IMPL>
                      <FORALL>
                        <VAR id="x" />
                        <PREDVAR id="\phi">
                          <FUNVAR id="g">
                            <VAR id="z" />
                          </FUNVAR>
                          <VAR id="x" />
                        </PREDVAR>
                      </FORALL>
                      <EXISTS>
                        <VAR id="y" />
                        <PREDVAR id="\phi">
                          <FUNVAR id="g">
                            <VAR id="z" />
                          </FUNVAR>
                          <VAR id="y" />
                        </PREDVAR>
                      </EXISTS>
                    </IMPL>
                  </FORMULA>
                  <!--  "FUNVAR" element is missing -->
                  <SUBST_FUNVAR ref="4">
                    <TERM>
                      <FUNVAR id="g">
                        <VAR id="u" />
                      </FUNVAR>
                    </TERM>
                  </SUBST_FUNVAR>
                </L>
              </LINES>
            </FORMAL_PROOF>
          </THEOREM>
        </NODE>
      </SUBSECTIONS>
    </SECTION>
  </CHAPTER>
</QEDEQ>
