<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> holds   <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">a</font> <a href="weddwitt.html#K2" title="WEDDWITT:func.2">-con_map</a> </span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> where <font color="Olive" title="b2">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Olive" title="b1">a</font> : verum </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Olive" title="b1">a</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="weddwitt.html#K2" title="WEDDWITT:func.2">-con_map</a> </span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font> : verum </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font></span>)</span></span><br/>



<b>reconsider </b><font color="Maroon" title="c3">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="weddwitt.html#K2" title="WEDDWITT:func.2">-con_map</a> </span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font> : verum </span>}</span>  as    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="weddwitt.html#T11" target="_self" title="WEDDWITT:th.11">Th11</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="weddwitt.html#K2" title="WEDDWITT:func.2">-con_map</a> </span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">$1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<a NAME="E2:18"/><i><font color="Green" title="E15">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 
;<br/>
<b>consider </b><font color="Maroon" title="c4">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">X</font><b> such that </b><br/><a NAME="E4:18"/><i><font color="Green" title="E16">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font> holds <br/><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S2" title="FUNCT_2:sch.2">FUNCT_2:sch 2</a>(<a class="txt" href="weddwitt.html#E2:18"><i><font color="Green" title="E15">A1</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon" title="c5">y</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font>;<br/>
<a NAME="E6:18"/>
for <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> implies  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )</span><br/>

<b>assume </b><a NAME="E1:18_1"/><i><font color="Green" title="E17">A3</font></i>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c7">c</font> = <font color="Maroon" title="c6">c</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="weddwitt.html#E1:18_1"><i><font color="Green" title="E17">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">y</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font><b> such that </b><br/><a NAME="E4:18_1"/><i><font color="Green" title="E18">A4</font></i>: 
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="weddwitt.html#K2" title="WEDDWITT:func.2">-con_map</a> </span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E1:18_1"><i><font color="Green" title="E17">A3</font></i></a></i>;<br/>
<a NAME="E5:18_1"/>
<font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E4:18"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="weddwitt.html#E4:18_1"><i><font color="Green" title="E18">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:18_1"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E7:18"/><b>then </b><i><font color="Green" title="E17">A5</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T16" title="FUNCT_2:th.16">FUNCT_2:16</a></i>;<br/>
<a NAME="E8:18"/><i><font color="Green" title="E18">A6</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:18"/>
for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">F</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">F</font> &amp; <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">F</font> &amp; <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">F</font> &amp; <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x2</font> implies <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:18_2"/><i><font color="Green" title="E19">A7</font></i>: 
<font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">F</font>
 <b>and </b><br/><a NAME="E2:18_2"/><i><font color="Green" title="E20">A8</font></i>: 
<font color="Maroon" title="c7">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">F</font>
 <b>and </b><br/><a NAME="E3:18_2"/><i><font color="Green" title="E21">A9</font></i>: 
<font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c8">y1</font> = <font color="Maroon" title="c6">x1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font> <b>by </b><i><a class="txt" href="weddwitt.html#E1:18_2"><i><font color="Green" title="E19">A7</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">y2</font> = <font color="Maroon" title="c7">x2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font> <b>by </b><i><a class="txt" href="weddwitt.html#E2:18_2"><i><font color="Green" title="E20">A8</font></i></a></i>;<br/>
<a NAME="E6:18_2"/><i><font color="Green" title="E22">A10</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="weddwitt.html#K2" title="WEDDWITT:func.2">-con_map</a> </span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="weddwitt.html#K2" title="WEDDWITT:func.2">-con_map</a> </span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">y2</font> )
 
<b>by </b><i><a class="txt" href="weddwitt.html#E4:18"><i><font color="Green" title="E16">A2</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1"/>
<font color="Maroon" title="c8">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">y2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:18_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="weddwitt.html#K2" title="WEDDWITT:func.2">-con_map</a> </span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="weddwitt.html#K2" title="WEDDWITT:func.2">-con_map</a> </span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="weddwitt.html#T10" target="_self" title="WEDDWITT:th.10">Th10</a></i>;<br/><a NAME="E3:18_2_1"/><b>then </b><i><font color="Green" title="E23">A11</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="weddwitt.html#K2" title="WEDDWITT:func.2">-con_map</a> </span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="weddwitt.html#K2" title="WEDDWITT:func.2">-con_map</a> </span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E5:18_2_1"/><i><font color="Green" title="E24">A12</font></i>: 
( <font color="Maroon" title="c10">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y1</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c10">d</font> <a href="group_3.html#R2" title="GROUP_3:pred.2">are_conjugated</a>  )
 <b>by </b><i><a class="ref" href="group_3.html#T95" title="GROUP_3:th.95">GROUP_3:95</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">g</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E7:18_2_1"/><i><font color="Green" title="E25">A13</font></i>: 
<font color="Maroon" title="c10">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c11">g</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E5:18_2_1"><i><font color="Green" title="E24">A12</font></i></a>, <a class="ref" href="group_3.html#D7" title="GROUP_3:def.7">GROUP_3:def 7</a></i>;<br/><a NAME="E8:18_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="weddwitt.html#K2" title="WEDDWITT:func.2">-con_map</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y1</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E5:18_2_1"><i><font color="Green" title="E24">A12</font></i></a>, <a class="txt" href="weddwitt.html#E7:18_2_1"><i><font color="Green" title="E25">A13</font></i></a>, <a class="ref" href="weddwitt.html#D2" target="_self" title="WEDDWITT:def.2">Def2</a></i>;<br/><a NAME="E9:18_2_1"/><b>then </b><i><font color="Green" title="E26">A14</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="weddwitt.html#K2" title="WEDDWITT:func.2">-con_map</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">y1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E10:18_2_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="weddwitt.html#K2" title="WEDDWITT:func.2">-con_map</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E11:18_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="weddwitt.html#E3:18_2"><i><font color="Green" title="E21">A9</font></i></a>, <a class="txt" href="weddwitt.html#E6:18_2"><i><font color="Green" title="E22">A10</font></i></a>, <a class="txt" href="weddwitt.html#E3:18_2_1"><i><font color="Green" title="E23">A11</font></i></a>, <a class="txt" href="weddwitt.html#E9:18_2_1"><i><font color="Green" title="E26">A14</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:18_2"/>
<font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E10:18"/><b>then </b>
<font color="Maroon" title="c4">F</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E11:18"/><b>then </b>
 <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="weddwitt.html#E8:18"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="card_1.html#T60" title="CARD_1:th.60">CARD_1:60</a></i>;<br/>
<a NAME="E12:18"/><b>then </b>
 <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font>, <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">F</font> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="weddwitt.html#E8:18"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="relat_1.html#T146" title="RELAT_1:th.146">RELAT_1:146</a></i>;<br/>
<b>hence </b><a NAME="E13:18"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="weddwitt.html#K2" title="WEDDWITT:func.2">-con_map</a> </span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font> : verum </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E7:18"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
