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<b>let </b><font color="Maroon">I</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">C</font> be   <a href="cat_1.html#NM4">Category</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">d</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_3.html#M2" target="_self">Injections_family</a> of <font color="Maroon">c</font>,<font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">F'</font> be    <a href="cat_3.html#M2" target="_self">Injections_family</a> of <font color="Maroon">d</font>,<font color="Maroon">I</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:90"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c</font> <a href="cat_3.html#R3" target="_self">is_a_coproduct_wrt</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E2:90"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">d</font> <a href="cat_3.html#R3" target="_self">is_a_coproduct_wrt</a> <font color="Maroon">F'</font>
 <b>and </b><br/><a NAME="E3:90"/><i><font color="Green">E43</font></i>: 
 <a href="cat_3.html#K3" target="_self">doms</a> <font color="Maroon">F</font> <a href="funct_2.html#R2">=</a>  <a href="cat_3.html#K3" target="_self">doms</a> <font color="Maroon">F'</font>
 ;<br/>

<b>consider </b><font color="Maroon">f</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E5:90"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>and </b><br/><a NAME="E6:90"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">k</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">d</font> holds <br/>( ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/><font color="Olive">k</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font> ) iff <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> )
 <b>by </b><i><a class="ref" href="cat_3.html#D3" target="_self">Def3</a>, <a class="ref" href="cat_3.html#T8" target="_self">Th8</a>, </i>;<br/>
<b>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="E8:90"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">d</font>,<font color="Maroon">c</font>
 <b>and </b><br/><a NAME="E9:90"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">k</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">d</font>,<font color="Maroon">c</font> holds <br/>( ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/><font color="Olive">k</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font> ) iff <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> )
 <b>by </b><i><a class="ref" href="cat_3.html#D4" target="_self">Def4</a>, <a class="ref" href="cat_3.html#T8" target="_self">Th8</a>, </i>;<br/>
<b>thus </b><a NAME="E10:90"/>
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">d</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D17">CAT_1:def 17</a><br/>

<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c</font>,<font color="Maroon">d</font> <b>by </b><i>, <a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>take </b>
<font color="Maroon">g</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D12">CAT_1:def 12</a><br/>

<a NAME="E12:90"/><i><font color="Green">E95</font></i>: 
(  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 
<b>by </b><i>, <a class="ref" href="cat_3.html#T10" target="_self">Th10</a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/>
<b>hence </b><a NAME="E13:90"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font> )
 ;<br/>

<a NAME="E14:90"/>
 <a href="cat_3.html#K3" target="_self">doms</a> <font color="Maroon">F'</font> <a href="funct_2.html#R2">=</a>  <a href="cat_3.html#K3" target="_self">doms</a> <font color="Maroon">F'</font>
 
;<br/>
<b>then consider </b><font color="Maroon">fg</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E16:90"/>
<font color="Maroon">fg</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">d</font>,<font color="Maroon">d</font>
 <b>and </b><br/><a NAME="E17:90"/><i><font color="Green">E97</font></i>: 
for <font color="Olive">k</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">d</font>,<font color="Maroon">d</font> holds <br/>( ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/><font color="Olive">k</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font> ) iff <font color="Maroon">fg</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> )
 <b>by </b><i><a class="ref" href="cat_3.html#D4" target="_self">Def4</a>, </i>;<br/>
<div><i><font color="Green">E98</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1"><b>now </b></a><div class="add"><b>set </b><font color="Maroon">k</font> =  <a href="cat_1.html#K10">id</a> <font color="Maroon">d</font>;<br/><b>thus </b><a NAME="E1:90_1"/>
 <a href="cat_1.html#K10">id</a> <font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">d</font>,<font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="cat_1.html#T55">CAT_1:55</a></i>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E2:90_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E3:90_1"/><b>then </b>
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="cat_3.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E4:90_1"/>
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">d</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="cat_1.html#T46">CAT_1:46</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_2"><b>now </b></a><div class="add"><a NAME="E1:90_2"/>
(  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp;  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> )
 <b>by </b><i>, <a class="ref" href="cat_1.html#T42">CAT_1:42</a></i>;<br/><b>hence </b><a NAME="E2:90_2"/>
<font color="Maroon">f</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">d</font>,<font color="Maroon">d</font>
 ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E3:90_2"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E4:90_2"/><b>then </b>
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="cat_3.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">g</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span> = 
<font color="Maroon">f</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="cat_1.html#T43">CAT_1:43</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_3.html#T10" target="_self">Th10</a>, , <a class="txt" href="cat_3.html#E8:90"><i><font color="Green">E62</font></i></a></i>
<br/>.= 
<font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="cat_3.html#T9" target="_self">Th9</a>, <a class="txt" href="cat_3.html#E8:90"><i><font color="Green">E62</font></i></a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><font color="Maroon">f</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">g</font> = 
<font color="Maroon">fg</font>
<b>by </b><i><a class="txt" href="cat_3.html#E5:90"><i><font color="Green">E60</font></i></a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i>, <a class="txt" href="cat_3.html#E5:90"><i><font color="Green">E60</font></i></a>, <a class="txt" href="cat_3.html#E6:90"><i><font color="Green">E61</font></i></a></i>
;<br/><br/>

<div><i><font color="Green">E100</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="90_4"><b>now </b></a><div class="add"><b>set </b><font color="Maroon">k</font> =  <a href="cat_1.html#K10">id</a> <font color="Maroon">c</font>;<br/><b>thus </b><a NAME="E1:90_4"/>
 <a href="cat_1.html#K10">id</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="cat_1.html#T55">CAT_1:55</a></i>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E2:90_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E3:90_4"/><b>then </b>
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="cat_3.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><a NAME="E4:90_4"/>
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="cat_1.html#T46">CAT_1:46</a></i>;<br/></div><b>end;</b></div>
<a NAME="E22:90"/>
 <a href="cat_3.html#K3" target="_self">doms</a> <font color="Maroon">F</font> <a href="funct_2.html#R2">=</a>  <a href="cat_3.html#K3" target="_self">doms</a> <font color="Maroon">F</font>
 
;<br/>
<b>then consider </b><font color="Maroon">gf</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E24:90"/>
<font color="Maroon">gf</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">c</font>
 <b>and </b><br/><a NAME="E25:90"/><i><font color="Green">E102</font></i>: 
for <font color="Olive">k</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">c</font> holds <br/>( ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/><font color="Olive">k</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font> ) iff <font color="Maroon">gf</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> )
 <b>by </b><i><a class="ref" href="cat_3.html#D3" target="_self">Def3</a>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_5"><b>now </b></a><div class="add"><a NAME="E1:90_5"/>
(  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp;  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 <b>by </b><i>, <a class="ref" href="cat_1.html#T42">CAT_1:42</a></i>;<br/><b>hence </b><a NAME="E2:90_5"/>
<font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">c</font>
 ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E3:90_5"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E4:90_5"/><b>then </b>
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="cat_3.html#T1" target="_self">Th1</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span> = 
<font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cat_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="cat_1.html#T43">CAT_1:43</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="cat_3.html#T9" target="_self">Th9</a>, <a class="ref" href="cat_3.html#D5" target="_self">Def5</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="cat_3.html#T10" target="_self">Th10</a>, , <a class="ref" href="cat_3.html#D5" target="_self">Def5</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font> = 
<font color="Maroon">gf</font>
<b>by </b><i><a class="ref" href="cat_3.html#T11" target="_self">Th11</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i>, <a class="txt" href="cat_3.html#E8:90"><i><font color="Green">E62</font></i></a>, <a class="ref" href="cat_3.html#T11" target="_self">Th11</a></i>
;<br/><br/>


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