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<b>let </b><font color="Maroon">C</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">B</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">C</font>,<font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">f'</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">C</font>,<font color="Maroon">D</font>;<br/>











<b>assume </b><a NAME="E1:8"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">F</font> is <a href="binop_1.html#V1">commutative</a> &amp; <font color="Maroon">F</font> is <a href="binop_1.html#V2">associative</a> &amp; ( <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">F</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  ) )
 ;<br/>

<b>set </b><font color="Maroon">s</font> =  <a href="partfun1.html#K6">id</a> <font color="Maroon">B</font>;<br/>
<b>assume </b><a NAME="E2:8"/>
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E3:8"/><b>then </b>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp;  <a href="partfun1.html#K6">id</a> <font color="Maroon">B</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">B</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T52">FUNCT_1:52</a>, <a class="ref" href="relat_1.html#T71">RELAT_1:71</a>, <a class="ref" href="relat_1.html#T94">RELAT_1:94</a></i>;<br/>
<b>hence </b><a NAME="E4:8"/>
<font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">B</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">B</font>,<font color="Maroon">f'</font>
 <b>by </b><i>, </i>;<br/>


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