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<b>let </b><font color="Maroon">C</font> be   <a href="cat_1.html#NM4">Category</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><a NAME="E1:53"/><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> 
 ;<br/>

<b>thus </b><a NAME="E2:53"/>
( <font color="Maroon">f</font> is <a href="cat_1.html#V4" target="_self">epi</a> implies for <font color="Olive">c</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font><br/> for <font color="Olive">g1</font>, <font color="Olive">g2</font> being   <a href="cat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">b</font>,<font color="Olive">c</font>  st  <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">b</font>,<font color="Olive">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">g1</font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">g2</font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">f</font> holds <br/><font color="Olive">g1</font> <a href="hidden.html#R1">=</a> <font color="Olive">g2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_1"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">g1</font>, <font color="Olive">g2</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>  st  <a href="cat_1.html#K2" target="_self">dom</a> <font color="Olive">g1</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3" target="_self">cod</a> <font color="Maroon">f</font> &amp;  <a href="cat_1.html#K2" target="_self">dom</a> <font color="Olive">g2</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3" target="_self">cod</a> <font color="Maroon">f</font> &amp;  <a href="cat_1.html#K3" target="_self">cod</a> <font color="Olive">g1</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3" target="_self">cod</a> <font color="Olive">g2</font> &amp; <font color="Olive">g1</font> <a href="cat_1.html#K4" target="_self">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">g2</font> <a href="cat_1.html#K4" target="_self">*</a> <font color="Maroon">f</font> holds <br/><font color="Olive">g1</font> <a href="hidden.html#R1">=</a> <font color="Olive">g2</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D11" target="_self">CAT_1:def 11</a><br/>

<b>let </b><font color="Maroon">c</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">g1</font> be    <a href="cat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">b</font>,<font color="Maroon">c</font>, <font color="Maroon">g2</font> be    <a href="cat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">b</font>,<font color="Maroon">c</font>;<br/>



<b>assume </b><a NAME="E2:53_1"/>
 <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E3:53_1"/><b>then </b>
(  <a href="cat_1.html#K2" target="_self">dom</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp;  <a href="cat_1.html#K2" target="_self">dom</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp;  <a href="cat_1.html#K3" target="_self">cod</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp;  <a href="cat_1.html#K3" target="_self">cod</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp;  <a href="cat_1.html#K3" target="_self">cod</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">g1</font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="cat_1.html#K4" target="_self">*</a> <font color="Maroon">f</font> &amp; <font color="Maroon">g2</font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="cat_1.html#K4" target="_self">*</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T18" target="_self">Th18</a>, <a class="ref" href="cat_1.html#T9" target="_self">Th9</a>, </i>;<br/>
<b>hence </b><a NAME="E4:53_1"/>
(  not <font color="Maroon">g1</font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">f</font> or <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> )
 <b>by </b><i><a class="ref" href="cat_1.html#D7" target="_self">Def7</a></i>;<br/>


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

<b>assume </b><a NAME="E3:53"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">c</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font><br/> for <font color="Olive">g1</font>, <font color="Olive">g2</font> being   <a href="cat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">b</font>,<font color="Olive">c</font>  st  <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">b</font>,<font color="Olive">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">g1</font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">g2</font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">f</font> holds <br/><font color="Olive">g1</font> <a href="hidden.html#R1">=</a> <font color="Olive">g2</font>
 ;<br/>

<b>let </b><font color="Maroon">g1</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>, <font color="Maroon">g2</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D11" target="_self">CAT_1:def 11</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E4:53"/><i><font color="Green">E23</font></i>: 
 <a href="cat_1.html#K2" target="_self">dom</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3" target="_self">cod</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E5:53"/><i><font color="Green">E50</font></i>: 
 <a href="cat_1.html#K2" target="_self">dom</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3" target="_self">cod</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E6:53"/><i><font color="Green">E51</font></i>: 
 <a href="cat_1.html#K3" target="_self">cod</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3" target="_self">cod</a> <font color="Maroon">g2</font>
 ;<br/>

<a NAME="E7:53"/><i><font color="Green">E52</font></i>: 
 <a href="cat_1.html#K3" target="_self">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="cat_1.html#T18" target="_self">Th18</a>, </i>;<br/>
<b>set </b><font color="Maroon">c</font> =  <a href="cat_1.html#K3" target="_self">cod</a> <font color="Maroon">g1</font>;<br/>
<b>reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">g1</font> as    <a href="cat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">b</font>, <a href="cat_1.html#K3" target="_self">cod</a> <font color="Maroon">g1</font> <b>by </b><i><a class="ref" href="cat_1.html#T22" target="_self">Th22</a>, <a class="ref" href="cat_1.html#T25" target="_self">Th25</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">g2</font> = <font color="Maroon">g2</font> as    <a href="cat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">b</font>, <a href="cat_1.html#K3" target="_self">cod</a> <font color="Maroon">g1</font> <b>by </b><i><a class="ref" href="cat_1.html#T23" target="_self">Th23</a>, , <a class="ref" href="cat_1.html#T25" target="_self">Th25</a>, </i>;<br/>
<a NAME="E10:53"/><i><font color="Green">E53</font></i>: 
 <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">b</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3" target="_self">cod</a> <font color="Maroon">g1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="cat_1.html#T22" target="_self">Th22</a>, <a class="ref" href="cat_1.html#T25" target="_self">Th25</a>, </i>;<br/>
<a NAME="E11:53"/><b>then </b>
( <font color="Maroon">g1</font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="cat_1.html#K4" target="_self">*</a> <font color="Maroon">f</font> &amp; <font color="Maroon">g2</font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="cat_1.html#K4" target="_self">*</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T18" target="_self">Th18</a>, <a class="ref" href="cat_1.html#T9" target="_self">Th9</a></i>;<br/>
<b>hence </b><a NAME="E12:53"/>
(  not <font color="Maroon">g1</font> <a href="cat_1.html#K4" target="_self">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="cat_1.html#K4" target="_self">*</a> <font color="Maroon">f</font> or <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> )
 <b>by </b><i><a class="ref" href="cat_1.html#D7" target="_self">Def7</a>, <a class="ref" href="cat_1.html#T26" target="_self">Th26</a></i>;<br/>


</div>
