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

<b>let </b><font color="Maroon">G</font> be  <a href="group_1.html#V6">finite</a> <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/>



<b>reconsider </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="weddwitt.html#K2" target="_self">-con_map</a> </span>)</span> <a href="relset_1.html#K11">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="group_3.html#K7">con_class</a> <font color="Maroon">a</font> : verum </span>}</span>  as    <a href="eqrel_1.html#M1">a_partition</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> <b>by </b><i/>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> = <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="weddwitt.html#K2" target="_self">-con_map</a> </span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>;<br/>
<a NAME="E2:18"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K7">con_class</a> <font color="Maroon">a</font> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
;<br/>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="group_3.html#K7">con_class</a> <font color="Maroon">a</font>,<font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:18"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K7">con_class</a> <font color="Maroon">a</font> holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S2">FUNCT_2:sch 2</a>();<br/></i>
<b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="group_3.html#K7">con_class</a> <font color="Maroon">a</font>;<br/>
<a NAME="E6:18"/>
for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/> ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K7">con_class</a> <font color="Maroon">a</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">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">c</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:18_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c</font> = <font color="Maroon">c</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="group_3.html#K7">con_class</a> <font color="Maroon">a</font><b> such that </b><br/><a NAME="E4:18_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="weddwitt.html#K2" target="_self">-con_map</a> </span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i/>;<br/>
<a NAME="E5:18_1"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:18_1"/>
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K7">con_class</a> <font color="Maroon">a</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:18"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T16">FUNCT_2:16</a></i>;<br/>
<a NAME="E8:18"/><i><font color="Green">E50</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="group_3.html#K7">con_class</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:18"/>
for <font color="Olive">x1</font>, <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x2</font> holds <br/><font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Olive">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">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:18_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E2:18_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E3:18_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">x1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="group_3.html#K7">con_class</a> <font color="Maroon">a</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">y2</font> = <font color="Maroon">x2</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="group_3.html#K7">con_class</a> <font color="Maroon">a</font> <b>by </b><i/>;<br/>
<a NAME="E6:18_2"/><i><font color="Green">E55</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="weddwitt.html#K2" target="_self">-con_map</a> </span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y1</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="weddwitt.html#K2" target="_self">-con_map</a> </span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y2</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y2</font> )
 
<b>by </b><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">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y2</font>
 ;<br/><a NAME="E2:18_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="weddwitt.html#K2" target="_self">-con_map</a> </span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y1</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="weddwitt.html#K2" target="_self">-con_map</a> </span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y2</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i/>;<br/><a NAME="E3:18_2_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="weddwitt.html#K2" target="_self">-con_map</a> </span>)</span> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y1</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="weddwitt.html#K2" target="_self">-con_map</a> </span>)</span> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y2</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>consider </b><font color="Maroon">d</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E5:18_2_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">d</font> <a href="group_3.html#R2">are_conjugated</a>  )
 <b>by </b><i><a class="ref" href="group_3.html#T95">GROUP_3:95</a></i>;<br/><a NAME="E6:18_2_1"/>
<font color="Maroon">d</font>,<font color="Maroon">a</font> <a href="group_3.html#R2">are_conjugated</a> 
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E8:18_2_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_3.html#K2">|^</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="group_3.html#D7">GROUP_3:def 7</a></i>;<br/><a NAME="E9:18_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="weddwitt.html#K2" target="_self">-con_map</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E10:18_2_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="weddwitt.html#K2" target="_self">-con_map</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y1</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E11:18_2_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="weddwitt.html#K2" target="_self">-con_map</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E12:18_2_1"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:18_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:18"/><b>then </b>
<font color="Maroon">F</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E11:18"/><b>then </b>
 <a href="group_3.html#K7">con_class</a> <font color="Maroon">a</font>,<font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="group_3.html#K7">con_class</a> <font color="Maroon">a</font></span>)</span> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i>, <a class="ref" href="card_1.html#T60">CARD_1:60</a></i>;<br/>
<a NAME="E12:18"/><b>then </b>
 <a href="group_3.html#K7">con_class</a> <font color="Maroon">a</font>, <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T146">RELAT_1:146</a></i>;<br/>
<b>hence </b><a NAME="E13:18"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="weddwitt.html#K2" target="_self">-con_map</a> </span>)</span> <a href="relset_1.html#K11">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="group_3.html#K7">con_class</a> <font color="Maroon">a</font> : verum </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="group_3.html#K7">con_class</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/>


</div>
