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<b>let </b><font color="Maroon">c<sub>1</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>2</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="ens_1.html#K5" target="_self">id$</a> <span class="p1">(<span class="default"><a href="ens_1.html#K3" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="ens_1.html#K5" target="_self">id$</a> <span class="p1">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<a NAME="E1:25"/><i><font color="Green">E17</font></i>: 
( (  <a href="ens_1.html#K4" target="_self">cod</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="ens_1.html#K3" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) &amp; <font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a>  is   <a href="funct_2.html#NM1">Function</a> of  <a href="ens_1.html#K3" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font>, <a href="ens_1.html#K4" target="_self">cod</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="ens_1.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E2:25"/>
 <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K5" target="_self">id$</a> <span class="p2">(<span class="default"><a href="ens_1.html#K3" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K3" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="ens_1.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E3:25"/><b>then </b>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K3" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="ens_1.html#K5" target="_self">id$</a> <span class="p2">(<span class="default"><a href="ens_1.html#K3" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="ens_1.html#K3" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="ens_1.html#K3" target="_self">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K5" target="_self">id$</a> <span class="p2">(<span class="default"><a href="ens_1.html#K3" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K3" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ens_1.html#K6" target="_self">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K5" target="_self">id$</a> <span class="p3">(<span class="default"><a href="ens_1.html#K3" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K5" target="_self">id$</a> <span class="p3">(<span class="default"><a href="ens_1.html#K3" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> &amp;  <a href="ens_1.html#K3" target="_self">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ens_1.html#K6" target="_self">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K5" target="_self">id$</a> <span class="p3">(<span class="default"><a href="ens_1.html#K3" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K3" target="_self">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K5" target="_self">id$</a> <span class="p2">(<span class="default"><a href="ens_1.html#K3" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> &amp;  <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="ens_1.html#K6" target="_self">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K5" target="_self">id$</a> <span class="p3">(<span class="default"><a href="ens_1.html#K3" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K4" target="_self">cod</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a>  )
 
<b>by </b><i><a class="txt" href="ens_1.html#E1:25"><i><font color="Green">E17</font></i></a>, <a class="ref" href="ens_1.html#T11" target="_self">Th11</a>, <a class="ref" href="ens_1.html#T12" target="_self">Th12</a>, <a class="ref" href="relat_1.html#T78">RELAT_1:78</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:25"/>
<font color="Maroon">c<sub>2</sub></font> <a href="ens_1.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K5" target="_self">id$</a> <span class="p2">(<span class="default"><a href="ens_1.html#K3" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="ens_1.html#E15"><i><font color="Green">Lemma8</font></i></a></i>;<br/>

<a NAME="E5:25"/>
(  <a href="ens_1.html#K3" target="_self">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K5" target="_self">id$</a> <span class="p2">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K4" target="_self">cod</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="ens_1.html#K4" target="_self">cod</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="ens_1.html#E1:25"><i><font color="Green">E17</font></i></a>, <a class="ref" href="ens_1.html#T11" target="_self">Th11</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E6:25"/><b>then </b>
( <span class="p1">(<span class="default"><a href="ens_1.html#K5" target="_self">id$</a> <span class="p2">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K5" target="_self">id$</a> <span class="p2">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K4" target="_self">cod</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K5" target="_self">id$</a> <span class="p3">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="ens_1.html#K6" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K5" target="_self">id$</a> <span class="p3">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> &amp;  <a href="ens_1.html#K3" target="_self">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K5" target="_self">id$</a> <span class="p3">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="ens_1.html#K6" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K3" target="_self">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K5" target="_self">id$</a> <span class="p3">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="ens_1.html#K6" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K5" target="_self">id$</a> <span class="p2">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a>  )
 
<b>by </b><i><a class="ref" href="ens_1.html#T11" target="_self">Th11</a>, <a class="ref" href="ens_1.html#T12" target="_self">Th12</a>, <a class="ref" href="relat_1.html#T79">RELAT_1:79</a></i>;<br/>
<b>hence </b><a NAME="E7:25"/>
<span class="p1">(<span class="default"><a href="ens_1.html#K5" target="_self">id$</a> <span class="p2">(<span class="default"><a href="ens_1.html#K4" target="_self">cod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="ens_1.html#K6" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="ens_1.html#E15"><i><font color="Green">Lemma8</font></i></a></i>;<br/>


</div>
