<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">l</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="coh_sp.html#K5" target="_self">MapsC</a> <font color="Maroon">X</font>;<br/>



<b>set </b><font color="Maroon">i1</font> =  <a href="coh_sp.html#K8" target="_self">id$</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">l</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">i2</font> =  <a href="coh_sp.html#K8" target="_self">id$</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">l</font></span>)</span>;<br/>
<a NAME="E1:42"/><i><font color="Green">E17</font></i>: 
( (  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">l</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">l</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) &amp; <font color="Maroon">l</font> <a href="mcart_1.html#K2">`2</a>  is   <a href="funct_2.html#NM1">Function</a> of  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">l</font></span>)</span>, <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">l</font></span>)</span> &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a>  <a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">l</font> holds <br/><span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a>  <a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">l</font> ) )
 
<b>by </b><i><a class="txt" href="coh_sp.html#E1:42"><i><font color="Green">E17</font></i></a></i>;<br/>
<a NAME="E2:42"/>
 <a href="coh_sp.html#K7" target="_self">cod</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K8" target="_self">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">l</font>
 
<b>by </b><i><a class="ref" href="coh_sp.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E3:42"/><b>then </b>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">l</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="coh_sp.html#K8" target="_self">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">l</font></span>)</span></span>)</span> &amp;  <a href="coh_sp.html#K6" target="_self">dom</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K8" target="_self">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">l</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="coh_sp.html#K9" target="_self">*</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K8" target="_self">id$</a> <span class="p3">(<span class="default"><a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">l</font></span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="coh_sp.html#K8" target="_self">id$</a> <span class="p3">(<span class="default"><a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> &amp;  <a href="coh_sp.html#K6" target="_self">dom</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="coh_sp.html#K9" target="_self">*</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K8" target="_self">id$</a> <span class="p3">(<span class="default"><a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">l</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="coh_sp.html#K6" target="_self">dom</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K8" target="_self">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">l</font></span>)</span></span>)</span> &amp;  <a href="coh_sp.html#K7" target="_self">cod</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="coh_sp.html#K9" target="_self">*</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K8" target="_self">id$</a> <span class="p3">(<span class="default"><a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">l</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">l</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">l</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">l</font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="mcart_1.html#K2">`2</a>  )
 
<b>by </b><i><a class="ref" href="coh_sp.html#D3" target="_self">Def3</a>, , <a class="ref" href="coh_sp.html#T2" target="_self">Th2</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="E4:42"/>
<font color="Maroon">l</font> <a href="coh_sp.html#K9" target="_self">*</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K8" target="_self">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font>
 <b>by </b><i/>;<br/>

<a NAME="E5:42"/>
(  <a href="coh_sp.html#K6" target="_self">dom</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K8" target="_self">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">l</font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">l</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="coh_sp.html#D3" target="_self">Def3</a>, <a class="ref" href="coh_sp.html#T2" target="_self">Th2</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E6:42"/><b>then </b>
( <span class="p1">(<span class="default"><a href="coh_sp.html#K8" target="_self">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">l</font></span>)</span></span>)</span> &amp;  <a href="coh_sp.html#K7" target="_self">cod</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K8" target="_self">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">l</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="coh_sp.html#K8" target="_self">id$</a> <span class="p3">(<span class="default"><a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="coh_sp.html#K9" target="_self">*</a> <font color="Maroon">l</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="coh_sp.html#K8" target="_self">id$</a> <span class="p3">(<span class="default"><a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">l</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">l</font> <a href="mcart_1.html#K2">`2</a> </span>)</span> &amp;  <a href="coh_sp.html#K6" target="_self">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="coh_sp.html#K8" target="_self">id$</a> <span class="p3">(<span class="default"><a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="coh_sp.html#K9" target="_self">*</a> <font color="Maroon">l</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="coh_sp.html#K6" target="_self">dom</a> <font color="Maroon">l</font> &amp;  <a href="coh_sp.html#K7" target="_self">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="coh_sp.html#K8" target="_self">id$</a> <span class="p3">(<span class="default"><a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="coh_sp.html#K9" target="_self">*</a> <font color="Maroon">l</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="coh_sp.html#K7" target="_self">cod</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K8" target="_self">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">l</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="tarski.html#K3">union</a> <span class="p3">(<span class="default"><a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">l</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="mcart_1.html#K2">`2</a>  )
 
<b>by </b><i>, <a class="ref" href="coh_sp.html#T2" target="_self">Th2</a>, <a class="ref" href="relat_1.html#T79">RELAT_1:79</a></i>;<br/>
<b>hence </b><a NAME="E7:42"/>
<span class="p1">(<span class="default"><a href="coh_sp.html#K8" target="_self">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K7" target="_self">cod</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="coh_sp.html#K9" target="_self">*</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font>
 <b>by </b><i/>;<br/>


</div>
