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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">c</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">i1</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">i2</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>;<br/>







<b>set </b><font color="Maroon">a</font> =  <a href="cat_1.html#K2">dom</a> <font color="Maroon">i1</font>;<br/>
<b>set </b><font color="Maroon">b</font> =  <a href="cat_1.html#K2">dom</a> <font color="Maroon">i2</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:105"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c</font> <a href="cat_3.html#R4" target="_self">is_a_coproduct_wrt</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font>
 <b>and </b><br/><a NAME="E2:105"/><i><font color="Green">E42</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">i1</font> is <a href="cat_1.html#V7">initial</a>
 ;<br/>

<a NAME="E3:105"/><i><font color="Green">E43</font></i>: 
(  <a href="cat_1.html#K3">cod</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:105"/>
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">i2</font></span>)</span>,<font color="Maroon">c</font> <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#T19">CAT_1:19</a></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">i</font> = <font color="Maroon">i2</font> as    <a href="cat_1.html#M1">Morphism</a> of  <a href="cat_1.html#K2">dom</a> <font color="Maroon">i2</font>,<font color="Maroon">c</font> <b>by </b><i><a class="txt" href="cat_3.html#E9:105"><i><font color="Green">E60</font></i></a>, <a class="ref" href="cat_1.html#T22">CAT_1:22</a></i>;<br/>
<b>take </b>
<font color="Maroon">i</font>
;<br/>

<b>set </b><font color="Maroon">f</font> =  <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">i2</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">g</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> <font color="Maroon">i1</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">i2</font></span>)</span>;<br/>
<a NAME="E6:105"/>
(  <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> <font color="Maroon">i1</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">i2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">i2</font> &amp;  <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> <font color="Maroon">i1</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">i2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">i1</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:105"/><b>then </b>
(  <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">i2</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> <font color="Maroon">i2</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">i2</font></span>)</span> &amp;  <a href="cat_3.html#K11" target="_self">init</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">i1</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">i2</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> <font color="Maroon">i1</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">i2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T55">CAT_1:55</a></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="E9:105"/><i><font color="Green">E60</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">c</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">i2</font></span>)</span>
 <b>and </b><br/><a NAME="E10:105"/><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>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">i2</font></span>)</span> holds <br/>( ( <font color="Olive">k</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">i1</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> <font color="Maroon">i1</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">i2</font></span>)</span> &amp; <font color="Olive">k</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a>  <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">i2</font></span>)</span> ) iff <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> )
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">h</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D12">CAT_1:def 12</a><br/>

<b>thus </b><a NAME="E11:105"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">i</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E9:105"><i><font color="Green">E60</font></i></a>, <a class="txt" href="cat_3.html#E10:105"><i><font color="Green">E61</font></i></a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1"><b>now </b></a><div class="add"><a NAME="E1:105_1"/><i><font color="Green">E62</font></i>: 
(  <a href="cat_1.html#K3">cod</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">i2</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E10:105"><i><font color="Green">E61</font></i></a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/><a NAME="E2:105_1"/><b>then </b><i><font color="Green">E78</font></i>: 
(  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">h</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">i</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">h</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E9:105"><i><font color="Green">E60</font></i></a>, <a class="ref" href="cat_1.html#T42">CAT_1:42</a></i>;<br/><b>hence </b><a NAME="E3:105_1"/>
<font color="Maroon">i</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">h</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/><a NAME="E4:105_1"/>
(  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">h</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">i1</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"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">h</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">i1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">i1</font> )
 <b>by </b><i><a class="txt" href="cat_3.html#E9:105"><i><font color="Green">E60</font></i></a>, <a class="ref" href="cat_3.html#D5" target="_self">Def5</a>, <a class="ref" href="cat_1.html#T42">CAT_1:42</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">h</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">i1</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> <font color="Maroon">i1</font></span>)</span>,<font color="Maroon">c</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">i1</font>
<b>by </b><i>, <a class="txt" href="cat_3.html#E9:105"><i><font color="Green">E60</font></i></a>, </i>
;<br/><br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">h</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">i2</font> = 
<font color="Maroon">i</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">i2</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E9:105"><i><font color="Green">E60</font></i></a>, <a class="ref" href="cat_3.html#T11" target="_self">Th11</a>, <a class="ref" href="cat_1.html#T43">CAT_1:43</a></i>
<br/>.= 
<font color="Maroon">i</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="cat_3.html#E10:105"><i><font color="Green">E61</font></i></a>, <a class="txt" href="cat_3.html#E1:105_1"><i><font color="Green">E62</font></i></a></i>
<br/>.= 
<font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="cat_1.html#T47">CAT_1:47</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:105"/>
<font color="Maroon">i</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <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">i</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="cat_3.html#E9:105"><i><font color="Green">E60</font></i></a>, <a class="ref" href="cat_3.html#T10" target="_self">Th10</a></i>;<br/>

<b>thus </b><a NAME="E14:105"/>
 <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">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="cat_3.html#E10:105"><i><font color="Green">E61</font></i></a>, </i>;<br/>


</div>
