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

<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<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>;<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="hidden.html#M1" title="HIDDEN:mode.1">set</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/>
<b>consider </b><font color="Maroon" title="c3">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E2:143"/><i><font color="Green" title="E96">A1</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">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>and </b><br/><a NAME="E3:143"/><i><font color="Green" title="E97">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="c3">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_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></i>
<a NAME="E4:143"/><i><font color="Green" title="E98">A3</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_3.html#K8" title="GROUP_3:func.8">con_class</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="143_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:143_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_3.html#K8" title="GROUP_3:func.8">con_class</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="143_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:143_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:143_1_1"/><i><font color="Green" title="E99">A4</font></i>: 
( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">y</font> = <font color="Maroon" title="c5">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="group_3.html#E2:143"><i><font color="Green" title="E96">A1</font></i></a>, <a class="txt" href="group_3.html#E3:143_1_1"><i><font color="Green" title="E99">A4</font></i></a></i>;<br/>
<a NAME="E5:143_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="group_3.html#E2:143"><i><font color="Green" title="E96">A1</font></i></a>, <a class="txt" href="group_3.html#E3:143"><i><font color="Green" title="E97">A2</font></i></a>, <a class="txt" href="group_3.html#E3:143_1_1"><i><font color="Green" title="E99">A4</font></i></a></i>;<br/>
<a NAME="E6:143_1_1"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">d</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> where <font color="Olive" title="b1">d</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : <font color="Olive" title="b1">d</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> </span>}</span> 
 
<b>by </b><i><a class="txt" href="group_3.html#E2:143"><i><font color="Green" title="E96">A1</font></i></a>, <a class="txt" href="group_3.html#E3:143_1_1"><i><font color="Green" title="E99">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:143_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K8" title="GROUP_3:func.8">con_class</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="group_3.html#T118" target="_self" title="GROUP_3:th.118">Th118</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:143_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K8" title="GROUP_3:func.8">con_class</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>

<a NAME="E3:143_1"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : <font color="Olive" title="b1">b</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> </span>}</span> 
 
<b>by </b><i><a class="ref" href="group_3.html#T118" target="_self" title="GROUP_3:th.118">Th118</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">b</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="E5:143_1"/><i><font color="Green" title="E99">A5</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c5">b</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> )
 ;<br/>
<a NAME="E6:143_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="group_3.html#E3:143"><i><font color="Green" title="E97">A2</font></i></a>, <a class="txt" href="group_3.html#E5:143_1"><i><font color="Green" title="E99">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:143_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="group_3.html#E2:143"><i><font color="Green" title="E96">A1</font></i></a>, <a class="txt" href="group_3.html#E5:143_1"><i><font color="Green" title="E99">A5</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:143"/>
<font color="Maroon" title="c3">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="143_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a><br/><b>let </b><font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:143_2"/><i><font color="Green" title="E99">A6</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> )
 <b>and </b><br/><a NAME="E2:143_2"/><i><font color="Green" title="E100">A7</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font>
 ;<br/>

<a NAME="E3:143_2"/>
( <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="group_3.html#E2:143"><i><font color="Green" title="E96">A1</font></i></a>, <a class="txt" href="group_3.html#E3:143"><i><font color="Green" title="E97">A2</font></i></a>, <a class="txt" href="group_3.html#E1:143_2"><i><font color="Green" title="E99">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:143_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font>
 <b>by </b><i><a class="txt" href="group_3.html#E2:143_2"><i><font color="Green" title="E100">A7</font></i></a>, <a class="ref" href="zfmisc_1.html#T6" title="ZFMISC_1:th.6">ZFMISC_1:6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:143"/><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="group_3.html#K8" title="GROUP_3:func.8">con_class</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="group_3.html#E2:143"><i><font color="Green" title="E96">A1</font></i></a>, <a class="txt" href="group_3.html#E4:143"><i><font color="Green" title="E98">A3</font></i></a>, <a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a></i>;<br/>
<b>hence </b> <a href="card_1.html#K1" title="CARD_1:func.1">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> = 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="group_3.html#K8" title="GROUP_3:func.8">con_class</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></i>
<br/>.= 
 <a href="group_2.html#K16" title="GROUP_2:func.16">Index</a> <span class="p1">(<span class="default"><a href="group_3.html#K10" title="GROUP_3:func.10">Normalizator</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T155" target="_self" title="GROUP_3:th.155">Th155</a></i>
;<br/><br/>


</div>
