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<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font><br/> for <font color="Olive" title="b2">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> holds <br/> ( <font color="Olive" title="b2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_5.html#K12" title="CAT_5:func.12">Hom</a> <font color="Olive" title="b1">a</font> iff  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> )</span><br/><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> holds <br/> ( <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_5.html#K12" title="CAT_5:func.12">Hom</a> <font color="Maroon" title="c2">a</font> iff  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> )</span><br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_5.html#K12" title="CAT_5:func.12">Hom</a> <font color="Maroon" title="c2">a</font> iff  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> )</span><br/>





<a NAME="E1:48"/>
(  <a href="cat_5.html#K12" title="CAT_5:func.12">Hom</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c1">C</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">f</font> &amp; (  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> implies  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> ) &amp; (  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> implies  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) )
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:48"/>
( <font color="Maroon" title="c3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_5.html#K12" title="CAT_5:func.12">Hom</a> <font color="Maroon" title="c2">a</font> iff  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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