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



<a NAME="E1:151"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="monoid_0.html#K18" target="_self">GPerms</a> <font color="Maroon">X</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="E2:151"/><b>then </b>
( <font color="Maroon">H<sub>1</sub></font>( <a href="monoid_0.html#K16" target="_self">GFuncs</a> <font color="Maroon">X</font>) <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">X</font> &amp; ( <font color="Maroon">x</font> is   <a href="funct_2.html#NM2">Permutation</a> of <font color="Maroon">X</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<font color="Maroon">X</font> ) &amp; ( <font color="Maroon">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K18" target="_self">GPerms</a> <font color="Maroon">X</font></span>)</span> implies <font color="Maroon">x</font> is   <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>, <a class="ref" href="funct_2.html#T12">FUNCT_2:12</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:151"/>
( <font color="Maroon">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="monoid_0.html#K18" target="_self">GPerms</a> <font color="Maroon">X</font></span>)</span> iff <font color="Maroon">x</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/>


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