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<b>let </b><font color="Maroon" title="c5">a</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cat_5.html#K12" title="CAT_5:func.12">Hom</a> <font color="Maroon" title="c2">o</font>;<br/>

<b>take </b><font color="Maroon" title="c6">f</font> =  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c5">a</font></span>)</span>;<br/>

<b>thus </b><a NAME="E1:53_2"/>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">f</font> )
 <b>by </b><i><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a>, <a class="ref" href="cat_1.html#T47" title="CAT_1:th.47">CAT_1:47</a></i>;<br/>

<b>let </b><font color="Maroon" title="c7">b</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cat_5.html#K12" title="CAT_5:func.12">Hom</a> <font color="Maroon" title="c2">o</font>;<br/><b>let </b><font color="Maroon" title="c8">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:53_2_1"/>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c8">g</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c8">g</font> )
 ;<br/><a NAME="E2:53_2_1"/><b>then </b>
 <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c8">g</font>
 <b>by </b><i><a class="ref" href="cat_1.html#T42" title="CAT_1:th.42">CAT_1:42</a></i>;<br/><b>hence </b><a NAME="E3:53_2_1"/>
<font color="Maroon" title="c8">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font>
 <b>by </b><i><a class="ref" href="cat_1.html#T47" title="CAT_1:th.47">CAT_1:47</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E3:53_2"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c8">g</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c8">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> )
 <b>by </b><i><a class="ref" href="cat_1.html#T46" title="CAT_1:th.46">CAT_1:46</a></i>;<br/>


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