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<b>let </b><font color="Maroon">G1</font> be  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V1">strict</a> <a href="group_1.html#V2">unital</a> <a href="monoid_0.html#V13" target="_self">invertible</a>  <a href="monoid_0.html#M5" target="_self">SubStr</a> of  <a href="monoid_0.html#K16" target="_self">GFuncs</a> <font color="Maroon">X</font>, <font color="Maroon">G2</font> be  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V1">strict</a> <a href="group_1.html#V2">unital</a> <a href="monoid_0.html#V13" target="_self">invertible</a>  <a href="monoid_0.html#M5" target="_self">SubStr</a> of  <a href="monoid_0.html#K16" target="_self">GFuncs</a> <font color="Maroon">X</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:150_1_2"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">f</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K16" target="_self">GFuncs</a> <font color="Maroon">X</font></span>)</span> holds <br/> ( <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">G1</font>) iff <font color="Olive">f</font> is   <a href="funct_2.html#NM2">Permutation</a> of <font color="Maroon">X</font> )
 <b>and </b><br/><a NAME="E2:150_1_2"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">f</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K16" target="_self">GFuncs</a> <font color="Maroon">X</font></span>)</span> holds <br/> ( <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">G2</font>) iff <font color="Olive">f</font> is   <a href="funct_2.html#NM2">Permutation</a> of <font color="Maroon">X</font> )
 ;<br/>

<a NAME="E3:150_1_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">G1</font>) <a href="tarski.html#R1">c=</a> <font color="Maroon">H<sub>1</sub></font>( <a href="monoid_0.html#K16" target="_self">GFuncs</a> <font color="Maroon">X</font>) &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">G2</font>) <a href="tarski.html#R1">c=</a> <font color="Maroon">H<sub>1</sub></font>( <a href="monoid_0.html#K16" target="_self">GFuncs</a> <font color="Maroon">X</font>) )
 
<b>by </b><i/>;<br/>
<a NAME="E4:150_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">G1</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">G2</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="150_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:150_1_2_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">G1</font>) <a href="tarski.html#R1">c=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">G2</font>)
  <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="150_1_2_1_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="E1:150_1_2_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">G1</font>)
 ;<br/>

<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K16" target="_self">GFuncs</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E3:150_1_2_1_1"/>
<font color="Maroon">f</font> is   <a href="funct_2.html#NM2">Permutation</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="txt" href="monoid_0.html#E41"><i><font color="Green">Lemma70</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E4:150_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">G2</font>)
 <b>by </b><i><a class="txt" href="monoid_0.html#E42"><i><font color="Green">Lemma71</font></i></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="E2:150_1_2_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">G2</font>)
 ;<br/>

<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K16" target="_self">GFuncs</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E4:150_1_2_1"/>
<font color="Maroon">f</font> is   <a href="funct_2.html#NM2">Permutation</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="txt" href="monoid_0.html#E42"><i><font color="Green">Lemma71</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E5:150_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">G1</font>)
 <b>by </b><i><a class="txt" href="monoid_0.html#E41"><i><font color="Green">Lemma70</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:150_1_2"/>
<font color="Maroon">G1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font>
 <b>by </b><i><a class="ref" href="monoid_0.html#T1" target="_self">Th1</a></i>;<br/>


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