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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">a</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">a</font>,<font color="Maroon">I</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:95"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">a</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="E2:95"/><i><font color="Green">E42</font></i>: 
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/> <a href="cat_1.html#K2">dom</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> is <a href="cat_1.html#V7">initial</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:95_1"/>
( <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">a</font> is <a href="cat_1.html#V7">initial</a> )
 <b>by </b><i>, <a class="ref" href="cat_3.html#T8" target="_self">Th8</a></i>;<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>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="cat_1.html#M1">Morphism</a> of  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">b</font> =  <a href="cat_3.html#K11" target="_self">init</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">b</font>;<br/><b>consider </b><font color="Maroon">F'</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">I</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:95_1"/><i><font color="Green">E43</font></i>: 
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="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="cat_3.html#S1" target="_self">CAT_3:sch 1</a>();<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:95_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E2:95_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="cat_1.html#K2">dom</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> is <a href="cat_1.html#V7">initial</a>
 <b>by </b><i><a class="ref" href="cat_3.html#T9" target="_self">Th9</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="cat_3.html#K4" target="_self">cods</a> <font color="Maroon">F'</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font> = 
 <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>
<b>by </b><i>, </i>
<br/>.= 
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><a href="cat_3.html#K11" target="_self">init</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span></span>)</span>,<font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_3.html#T10" target="_self">Th10</a>, </i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="cat_3.html#K1" target="_self">--&gt;</a> <font color="Maroon">b</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div><a NAME="E5:95_1"/><b>then </b>
 <a href="cat_3.html#K4" target="_self">cods</a> <font color="Maroon">F'</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">I</font> <a href="cat_3.html#K1" target="_self">--&gt;</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">F'</font> = <font color="Maroon">F'</font> as    <a href="cat_3.html#M2" target="_self">Injections_family</a> of <font color="Maroon">b</font>,<font color="Maroon">I</font> <b>by </b><i><a class="txt" href="cat_3.html#E3:95_1"><i><font color="Green">E43</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:95_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E2:95_1_2"/><b>then </b><i><font color="Green">E78</font></i>: 
 <a href="cat_1.html#K2">dom</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> is <a href="cat_1.html#V7">initial</a>
 <b>by </b><i><a class="ref" href="cat_3.html#T9" target="_self">Th9</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="cat_3.html#K3" target="_self">doms</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font> = 
 <a href="cat_1.html#K2">dom</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>, </i>
<br/>.= 
 <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="cat_3.html#K11" target="_self">init</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span></span>)</span>,<font color="Maroon">b</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="cat_1.html#K2">dom</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>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_3.html#K3" target="_self">doms</a> <font color="Maroon">F'</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div><a NAME="E8:95_1"/><b>then </b>
 <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>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">h</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E10:95_1"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E11:95_1"/><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">a</font>,<font color="Maroon">b</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">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> )
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E12:95_1"/>
 <a href="cat_1.html#K6">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> 
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">h</font> as    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</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">h</font> = <font color="Maroon">h</font>;<br/><b>let </b><font color="Maroon">g</font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1_3"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:95_1_3"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>set </b><font color="Maroon">c</font> =  <a href="cat_1.html#K2">dom</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>;<br/><b>assume </b><a NAME="E2:95_1_3"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E3:95_1_3"/><b>then </b>
 <a href="cat_1.html#K2">dom</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> is <a href="cat_1.html#V7">initial</a>
 <b>by </b><i><a class="ref" href="cat_3.html#T9" target="_self">Th9</a></i>;<br/><b>then consider </b><font color="Maroon">f</font> being    <a href="cat_1.html#M1">Morphism</a> of  <a href="cat_1.html#K2">dom</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">b</font><b> such that </b><br/><a NAME="E5:95_1_3"/><i><font color="Green">E99</font></i>: 
for <font color="Olive">f'</font> being   <a href="cat_1.html#M1">Morphism</a> of  <a href="cat_1.html#K2">dom</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">b</font> holds  <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">f'</font>
 <b>by </b><i><a class="ref" href="cat_1.html#D16">CAT_1:def 16</a></i>;<br/><a NAME="E6:95_1_3"/>
(  <a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <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">a</font> )
 <b>by </b><i>, , <a class="ref" href="cat_3.html#T1" target="_self">Th1</a>, <a class="ref" href="cat_1.html#T23">CAT_1:23</a></i>;<br/><a NAME="E7:95_1_3"/><b>then </b>
(  <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> <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> <a href="hidden.html#R1">=</a> <font color="Maroon">b</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> <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> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</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_1.html#T42">CAT_1:42</a></i>;<br/><a NAME="E8:95_1_3"/><b>then </b>
<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> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</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>,<font color="Maroon">b</font>
 ;<br/><a NAME="E9:95_1_3"/><b>then </b><i><font color="Green">E100</font></i>: 
<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> is    <a href="cat_1.html#M1">Morphism</a> of  <a href="cat_1.html#K2">dom</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">b</font>
 <b>by </b><i><a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/><a NAME="E10:95_1_3"/>
<font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="cat_3.html#K11" target="_self">init</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</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>,<font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="cat_3.html#T10" target="_self">Th10</a>, </i>;<br/><b>hence </b><font color="Maroon">F'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font> = 
<font color="Maroon">f</font>
<b>by </b><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>, </i>
;<br/><br/></div><b>end;</b></div><b>hence </b><a NAME="E15:95_1"/>
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:95"/>
<font color="Maroon">a</font> is <a href="cat_1.html#V7">initial</a>
 <b>by </b><i><a class="ref" href="cat_1.html#D16">CAT_1:def 16</a></i>;<br/>


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