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<b>let </b><font color="Maroon">C</font> be    <a href="cat_1.html#L1" target="_self">CatStr</a> ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="cat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:18"/><i><font color="Green">E20</font></i>: 
 <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:18"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">g</font> being   <a href="cat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font> holds  <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font>
 ;<br/>

<a NAME="E3:18"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> iff <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </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="E1:18_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> implies <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 ;<br/>

<b>then consider </b><font color="Maroon">g</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:18_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E4:18_1_1"/><i><font color="Green">E23</font></i>: 
(  <a href="cat_1.html#K2" target="_self">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="cat_1.html#K3" target="_self">cod</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 ;<br/>
<a NAME="E5:18_1_1"/>
<font color="Maroon">g</font> is    <a href="cat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:18_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="cat_1.html#T4" target="_self">Th4</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:18_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:18"/>
 <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">f</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


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