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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>;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:71"/><i><font color="Green" title="E43">A1</font></i>: 
<font color="Maroon" title="c2">a</font> is <a href="cat_1.html#V7" title="CAT_1:attr.7">initial</a>
 <b>and </b><br/><a NAME="E2:71"/><i><font color="Green" title="E44">A2</font></i>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="cat_1.html#R1" title="CAT_1:pred.1">are_isomorphic</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c4">c</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D16" target="_self" title="CAT_1:def.16">CAT_1:def 16</a><br/>

<b>consider </b><font color="Maroon" title="c5">h</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font><b> such that </b><br/><a NAME="E4:71"/><i><font color="Green" title="E45">A3</font></i>: 
for <font color="Olive" title="b1">g</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> holds  <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font>
 <b>by </b><i><a class="txt" href="cat_1.html#E1:71"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="cat_1.html#D16" target="_self" title="CAT_1:def.16">Def16</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">f</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c7">f'</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font><b> such that </b><br/><a NAME="E6:71"/><i><font color="Green" title="E46">A4</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c7">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">b</font>
 <b>and </b><br/><a NAME="E7:71"/>
<font color="Maroon" title="c7">f'</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="cat_1.html#E2:71"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="cat_1.html#T81" target="_self" title="CAT_1:th.81">Th81</a></i>;<br/>
<a NAME="E8:71"/><i><font color="Green" title="E47">A5</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="cat_1.html#E2:71"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="cat_1.html#T81" target="_self" title="CAT_1:th.81">Th81</a></i>;<br/>
<a NAME="E9:71"/>
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="cat_1.html#E1:71"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="cat_1.html#D16" target="_self" title="CAT_1:def.16">Def16</a></i>;<br/>
<b>hence </b><a NAME="E10:71"/><i><font color="Green" title="E48">A6</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="cat_1.html#E8:71"><i><font color="Green" title="E47">A5</font></i></a>, <a class="ref" href="cat_1.html#T51" target="_self" title="CAT_1:th.51">Th51</a></i>;<br/>

<b>take </b>
<font color="Maroon" title="c5">h</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c7">f'</font>
;<br/>

<a NAME="E11:71"/><i><font color="Green" title="E49">A7</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="cat_1.html#E2:71"><i><font color="Green" title="E44">A2</font></i></a>, <a class="ref" href="cat_1.html#D17" target="_self" title="CAT_1:def.17">Def17</a></i>;<br/>
<b>let </b><font color="Maroon" title="c8">h'</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>;<br/>

<b>thus </b><font color="Maroon" title="c8">h'</font> = 
<font color="Maroon" title="c8">h'</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c7">f'</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_1.html#E6:71"><i><font color="Green" title="E46">A4</font></i></a>, <a class="txt" href="cat_1.html#E10:71"><i><font color="Green" title="E48">A6</font></i></a>, <a class="ref" href="cat_1.html#T58" target="_self" title="CAT_1:th.58">Th58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">h'</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c7">f'</font>
<b>by </b><i><a class="txt" href="cat_1.html#E8:71"><i><font color="Green" title="E47">A5</font></i></a>, <a class="txt" href="cat_1.html#E10:71"><i><font color="Green" title="E48">A6</font></i></a>, <a class="txt" href="cat_1.html#E11:71"><i><font color="Green" title="E49">A7</font></i></a>, <a class="ref" href="cat_1.html#T54" target="_self" title="CAT_1:th.54">Th54</a></i>
<br/>.= 
<font color="Maroon" title="c5">h</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c7">f'</font>
<b>by </b><i><a class="txt" href="cat_1.html#E4:71"><i><font color="Green" title="E45">A3</font></i></a></i>
;<br/>


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