<?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_07/xml/qedeq.xsd">
  <HEADER email="mime@qedeq.org">
    <SPECIFICATION name="MCEC072" ruleVersion="1.00.00">
      <LOCATIONS>
        <LOCATION value="file:///."/>
      </LOCATIONS>
    </SPECIFICATION>
    <TITLE>
      <LATEX language="en">
         Class Operator Definition
      </LATEX>
    </TITLE>
    <ABSTRACT>
      <LATEX language="en">
         We define the class operator constant and some other things.
      </LATEX>
    </ABSTRACT>
    <AUTHORS>
      <AUTHOR email="michael@meyling.com">
        <NAME>
          <LATEX language="de">
            Michael Meyling
          </LATEX>
        </NAME>
      </AUTHOR>
    </AUTHORS>
  </HEADER>
  <CHAPTER>
    <TITLE>
      <LATEX language="en">
        The Only One
      </LATEX>
    </TITLE>
    <INTRODUCTION>
      <LATEX language="en">
        <![CDATA[
          This module is for testing purposes only.
        ]]>
      </LATEX>
    </INTRODUCTION>
    <SECTION>
      <TITLE>
        <LATEX language="en">Identity</LATEX>
      </TITLE>
      <INTRODUCTION>
        <LATEX language="en">
          <![CDATA[
            We have a definition some axioms and a proposition.
          ]]>
        </LATEX>
      </INTRODUCTION>
      <SUBSECTIONS>
        <NODE id="definition:in">
          <NAME>
            <LATEX language="en">is a member of</LATEX>
            <LATEX language="de">ist enthalten in</LATEX>
          </NAME>
          <TITLE>
            <LATEX language="en">Membership Operator</LATEX>
            <LATEX language="de">Elementbeziehung</LATEX>
          </TITLE>
          <PRECEDING>
            <LATEX language="en">
              <![CDATA[
                The theory of sets introduced here has initial objects, called \emph{classes}. Furthermore the only predefined symbol is for a binary relation called \emph{membership}.
              ]]>
            </LATEX>
            <LATEX language="de">
              <![CDATA[
                Die hier vorgestellte Mengenlehre hat als Ausgangsobjekte \emph{Klassen}.
                Weiterhin wird nur ein einziges Symbol für eine binäre Relation vorausgesetzt: der \emph{Enthaltenseinoperator}.
              ]]>
            </LATEX>
          </PRECEDING>
          <DEFINITION_PREDICATE_INITIAL arguments="2" name="in">
            <LATEXPATTERN>#1 \in #2</LATEXPATTERN>
            <PREDCON ref="in">
              <VAR id="x"/>
              <VAR id="y"/>
            </PREDCON>
          </DEFINITION_PREDICATE_INITIAL>
          <SUCCEEDING>
            <LATEX language="en">
              <![CDATA[
                We also say ``$x$ \emph{is element of} $y$'', ``$x$ \emph{belongs to} $y$'', ``$x$ \emph{is a member of} $y$'' or ``$x$ \emph{is in} $y$''.
                Beside identity this is the only predicate we start with. All other will be defined. Also no function constants are initially given. Later on we will successively fix their meaning.
              ]]>
            </LATEX>
          </SUCCEEDING>
        </NODE>

        <NODE id="rule:classDefinition">
          <NAME>
            <LATEX language="en">class definition</LATEX>
          </NAME>
          <TITLE>
            <LATEX language="en">Class definition</LATEX>
          </TITLE>
          <PRECEDING>
            <LATEX language="en">
              <![CDATA[
                Starting with \qref{theorem:comprehension} we can extend the syntax and provide a new abbreviation.
              ]]>
            </LATEX>
          </PRECEDING>
          <RULE name="CLASS_DEFINITION_BY_FORMULA" version="1.00.00">
            <LINK id="theorem:comprehension"/>
            <DESCRIPTION>
              <LATEX language="en">
                <![CDATA[
                  For every formula $\alpha(x_1)$ we define the term expression $\{ x_1 \ | \ \alpha(x_1)\}$ where $x_1$ is a subject variable that is not bound within $\alpha(x_1)$. The free variables of $\{ x_1 \ | \ \alpha(x_1)\}$ are the are the free variables of $\alpha(x_1)$. The bound variables correspond to the bound variables of $\alpha(x_1)$ enhanced by $x_1$. 
                  \par
                  All deduction rules are accordingly extended.
                  
                ]]>
              </LATEX>
            </DESCRIPTION>
          </RULE>
          <SUCCEEDING>
            <LATEX language="en">
              <![CDATA[
                In particular the substitution rules must be adapted because now a term can have bound subject variables. \footnote{Luckily we formulated the substitution rules with this extension already in our mind, so we have nothing to do.}
                In the following this new notation is used.
              ]]>
            </LATEX>
          </SUCCEEDING>
        </NODE>
       
      </SUBSECTIONS>
    </SECTION>
  </CHAPTER>
</QEDEQ>
