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

<b>let </b><font color="Maroon">G</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">H</font> be  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">H1</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">H1</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Olive">H1</font> );<br/>
<a NAME="E1:147"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</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#K9" target="_self">con_class</a> <font color="Maroon">H</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
;<br/>
<a NAME="E2:147"/><i><font color="Green">E39</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#K9" target="_self">con_class</a> <font color="Maroon">H</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:147_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">H</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">H</font> = <font color="Maroon">x</font> as  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">y</font> =  <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font> as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>take </b>
<font color="Maroon">H</font>
;<br/>

<b>thus </b><a NAME="E4:147_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font> )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:147"/><i><font color="Green">E40</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">H</font>
 <b>and </b><br/><a NAME="E5:147"/><i><font color="Green">E49</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#K9" target="_self">con_class</a> <font color="Maroon">H</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(<a class="ref" href="group_3.html#T72" target="_self">Th72</a>, <a class="ref" href="group_3.html#T73" target="_self">Th73</a>);<br/></i>
<a NAME="E6:147"/><i><font color="Green">E50</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="group_3.html#K8" target="_self">con_class</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:147_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="group_3.html#K8" target="_self">con_class</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:147_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:147_2_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">H1</font> being  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E5:147_2_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H1</font> )
 <b>by </b><i><a class="ref" href="group_3.html#T74" target="_self">Th74</a>, <a class="ref" href="group_3.html#T76" target="_self">Th76</a>, <a class="ref" href="group_3.html#T78" target="_self">Th78</a></i>;<br/>
<a NAME="E6:147_2_1"/>
<font color="Maroon">H1</font>,<font color="Maroon">H</font> <a href="group_3.html#R6" target="_self">are_conjugated</a> 
 
<b>by </b><i><a class="ref" href="group_3.html#T74" target="_self">Th74</a>, <a class="ref" href="group_3.html#T78" target="_self">Th78</a>, <a class="ref" href="group_3.html#T79" target="_self">Th79</a>, <a class="ref" href="group_3.html#T39" target="_self">Th39</a></i>;<br/>
<a NAME="E7:147_2_1"/><b>then </b>
 <a href="group_2.html#K7">carr</a> <font color="Maroon">H1</font>, <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font> <a href="group_3.html#R4" target="_self">are_conjugated</a> 
 
<b>by </b><i><a class="ref" href="group_3.html#T42" target="_self">Th42</a></i>;<br/>
<b>hence </b><a NAME="E8:147_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K8" target="_self">con_class</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_3.html#T79" target="_self">Th79</a></i>;<br/>


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

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

<b>assume </b><a NAME="E2:147_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K8" target="_self">con_class</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">B</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E4:147_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font>,<font color="Maroon">B</font> <a href="group_3.html#R4" target="_self">are_conjugated</a>  )
 ;<br/>
<b>consider </b><font color="Maroon">H1</font> being  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E6:147_2"/><i><font color="Green">E54</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="group_3.html#T78" target="_self">Th78</a>, <a class="ref" href="group_3.html#T30" target="_self">Th30</a></i>;<br/>
<a NAME="E7:147_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H1</font>
 
<b>by </b><i><a class="ref" href="group_3.html#T79" target="_self">Th79</a></i>;<br/>
<a NAME="E8:147_2"/><b>then </b>
<font color="Maroon">H1</font>,<font color="Maroon">H</font> <a href="group_3.html#R6" target="_self">are_conjugated</a> 
 
<b>by </b><i><a class="ref" href="group_3.html#T78" target="_self">Th78</a>, <a class="ref" href="group_3.html#T42" target="_self">Th42</a></i>;<br/>
<a NAME="E9:147_2"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">H1</font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="group_3.html#T39" target="_self">Th39</a></i>;<br/>
<a NAME="E10:147_2"/><b>then </b>
 ex <font color="Olive">H2</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <br/>( <font color="Maroon">H1</font> <a href="group_2.html#R1">=</a> <font color="Olive">H2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">H1</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Olive">H2</font> )
 
<b>by </b><i><a class="ref" href="group_3.html#T76" target="_self">Th76</a></i>;<br/>
<b>hence </b><a NAME="E11:147_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="group_3.html#T74" target="_self">Th74</a>, <a class="ref" href="group_3.html#T78" target="_self">Th78</a>, <a class="ref" href="group_3.html#T80" target="_self">Th80</a>, <a class="ref" href="group_3.html#T82" target="_self">Th82</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:147"/>
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_3"><b>proof </b></a><div class="add">

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



<b>assume </b><b>that </b><br/><a NAME="E1:147_3"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E2:147_3"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E3:147_3"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 ;<br/>

<b>consider </b><font color="Maroon">H1</font> being  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E5:147_3"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H1</font> )
 <b>by </b><i><a class="ref" href="group_3.html#T74" target="_self">Th74</a>, <a class="ref" href="group_3.html#T76" target="_self">Th76</a>, <a class="ref" href="group_3.html#T78" target="_self">Th78</a></i>;<br/>
<b>consider </b><font color="Maroon">H2</font> being  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E7:147_3"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H2</font> )
 <b>by </b><i><a class="ref" href="group_3.html#T74" target="_self">Th74</a>, <a class="ref" href="group_3.html#T76" target="_self">Th76</a>, <a class="ref" href="group_3.html#T79" target="_self">Th79</a></i>;<br/>
<b>thus </b><a NAME="E8:147_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="group_3.html#T80" target="_self">Th80</a>, <a class="ref" href="group_3.html#T82" target="_self">Th82</a>, <a class="ref" href="group_3.html#T84" target="_self">Th84</a>, <a class="ref" href="group_2.html#T68">GROUP_2:68</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:147"/><b>then </b>
 <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">H</font>, <a href="group_3.html#K8" target="_self">con_class</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="group_3.html#T74" target="_self">Th74</a>, <a class="ref" href="group_3.html#T77" target="_self">Th77</a>, <a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/>
<b>hence </b> <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">H</font></span>)</span> = 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="group_3.html#K8" target="_self">con_class</a> <span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="card_1.html#T21">CARD_1:21</a></i>
<br/>.= 
 <a href="group_2.html#K16">Index</a> <span class="p1">(<span class="default"><a href="group_3.html#K11" target="_self">Normalizator</a> <font color="Maroon">H</font></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T65" target="_self">Th65</a></i>
;<br/><br/>


</div>
