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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>













<b>assume </b><b>that </b><br/><a NAME="E1:7"/><i><font color="Green">E6</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> is <a href="binop_1.html#V1">commutative</a> &amp; <font color="Maroon">c<sub>6</sub></font> is <a href="binop_1.html#V2">associative</a> &amp; ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>6</sub></font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  ) )
 <b>and </b><br/><a NAME="E2:7"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E3:7"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/>
<a NAME="E4:7"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:7"/><b>then </b>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="setwiseo.html#K8">.:</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="setwop_2.html#E2:7"><i><font color="Green">E7</font></i></a>, <a class="ref" href="funct_1.html#T84">FUNCT_1:84</a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a>, <a class="ref" href="relat_1.html#T112">RELAT_1:112</a>, <a class="ref" href="relat_1.html#T148">RELAT_1:148</a></i>;<br/>
<b>hence </b><a NAME="E6:7"/>
<font color="Maroon">c<sub>6</sub></font> <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="setwiseo.html#K8">.:</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="setwop_2.html#E1:7"><i><font color="Green">E6</font></i></a>, <a class="ref" href="setwop_2.html#T7" target="_self">Th7</a></i>;<br/>


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