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<b>let </b><font color="Maroon">V</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">a</font> be   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" target="_self">Ens</a> <font color="Maroon">V</font></span>)</span>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" target="_self">Ens</a> <font color="Maroon">V</font></span>)</span>;<br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:45_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ens_1.html#K7" target="_self">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">b</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_1_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">x</font> as   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" target="_self">Ens</a> <font color="Maroon">V</font></span>)</span> ;<br/>
<a NAME="E3:45_1_1"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="ens_1.html#T4" target="_self">Th4</a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/>
<a NAME="E4:45_1_1"/><b>then </b>
(  <a href="ens_1.html#K3" target="_self">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E5:45_1_1"/><b>then </b>
(  <a href="ens_1.html#K3" target="_self">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">a</font> &amp;  <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">b</font> )
 
;<br/>
<a NAME="E6:45_1_1"/><b>then </b>
 <a href="ens_1.html#K16" target="_self">@</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="ens_1.html#K7" target="_self">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:45_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ens_1.html#K7" target="_self">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">b</font></span>)</span>
 ;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:45_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ens_1.html#K7" target="_self">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">b</font></span>)</span>
 ;<br/><a NAME="E3:45_1"/>
 <a href="ens_1.html#K7" target="_self">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">b</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">m</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K2" target="_self">Maps</a> <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="ens_1.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E5:45_1"/>
(  <a href="ens_1.html#K3" target="_self">dom</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">a</font> &amp;  <a href="ens_1.html#K4" target="_self">cod</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="ens_1.html#T4" target="_self">Th4</a>, </i>;<br/><a NAME="E6:45_1"/><b>then </b>
(  <a href="ens_1.html#K3" target="_self">dom</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="ens_1.html#K4" target="_self">cod</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 ;<br/><a NAME="E7:45_1"/><b>then </b>
(  <a href="ens_1.html#K3" target="_self">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="ens_1.html#K4" target="_self">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" target="_self">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 ;<br/><a NAME="E8:45_1"/><b>then </b>
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/><a NAME="E9:45_1"/><b>then </b>
 <a href="ens_1.html#K15" target="_self">@</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 ;<br/><b>hence </b><a NAME="E10:45_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:45"/>
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="ens_1.html#K7" target="_self">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K14" target="_self">@</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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