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<b>let </b><font color="Maroon" title="c2">G1</font>, <font color="Maroon" title="c3">G2</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a> <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a> <a href="monoid_0.html#V13" title="MONOID_0:attr.13">invertible</a>  <a href="monoid_0.html#M5" title="MONOID_0:mode.5">SubStr</a> of  <a href="monoid_0.html#K15" title="MONOID_0:func.15">GFuncs</a> <font color="Maroon" title="c1">X</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:146_1_2"/><i><font color="Green" title="E66">A5</font></i>: 
for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K15" title="MONOID_0:func.15">GFuncs</a> <font color="Maroon" title="c1">X</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">G1</font>) iff <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c1">X</font> )
 <b>and </b><br/><a NAME="E2:146_1_2"/><i><font color="Green" title="E67">A6</font></i>: 
for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K15" title="MONOID_0:func.15">GFuncs</a> <font color="Maroon" title="c1">X</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">G2</font>) iff <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c1">X</font> )
 ;<br/>

<a NAME="E3:146_1_2"/><i><font color="Green" title="E68">A7</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">G1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">H<sub>1</sub></font>( <a href="monoid_0.html#K15" title="MONOID_0:func.15">GFuncs</a> <font color="Maroon" title="c1">X</font>) &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">G2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">H<sub>1</sub></font>( <a href="monoid_0.html#K15" title="MONOID_0:func.15">GFuncs</a> <font color="Maroon" title="c1">X</font>) )
 
<b>by </b><i><a class="ref" href="monoid_0.html#T25" target="_self" title="MONOID_0:th.25">Th25</a></i>;<br/>
<a NAME="E4:146_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">G1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">G2</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="146_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:146_1_2_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">G1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">G2</font>)
  <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="146_1_2_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="E1:146_1_2_1_1"/><i><font color="Green" title="E69">A8</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">G1</font>)
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K15" title="MONOID_0:func.15">GFuncs</a> <font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="txt" href="monoid_0.html#E3:146_1_2"><i><font color="Green" title="E68">A7</font></i></a></i>;<br/>
<a NAME="E3:146_1_2_1_1"/>
<font color="Maroon" title="c5">f</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="monoid_0.html#E1:146_1_2"><i><font color="Green" title="E66">A5</font></i></a>, <a class="txt" href="monoid_0.html#E1:146_1_2_1_1"><i><font color="Green" title="E69">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:146_1_2_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">G2</font>)
 <b>by </b><i><a class="txt" href="monoid_0.html#E2:146_1_2"><i><font color="Green" title="E67">A6</font></i></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="E2:146_1_2_1"/><i><font color="Green" title="E69">A9</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">G2</font>)
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K15" title="MONOID_0:func.15">GFuncs</a> <font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="txt" href="monoid_0.html#E3:146_1_2"><i><font color="Green" title="E68">A7</font></i></a></i>;<br/>
<a NAME="E4:146_1_2_1"/>
<font color="Maroon" title="c5">f</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="monoid_0.html#E2:146_1_2"><i><font color="Green" title="E67">A6</font></i></a>, <a class="txt" href="monoid_0.html#E2:146_1_2_1"><i><font color="Green" title="E69">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:146_1_2_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">G1</font>)
 <b>by </b><i><a class="txt" href="monoid_0.html#E1:146_1_2"><i><font color="Green" title="E66">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:146_1_2"/>
<font color="Maroon" title="c2">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G2</font>
 <b>by </b><i><a class="ref" href="monoid_0.html#T28" target="_self" title="MONOID_0:th.28">Th28</a></i>;<br/>


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