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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">F</font> be    <a href="cat_3.html#M2" target="_self">Injections_family</a> of <font color="Maroon">a</font>, <a href="xboole_0.html#K1">{}</a> ;<br/>





<b>thus </b><a NAME="E1:93"/>
( <font color="Maroon">a</font> <a href="cat_3.html#R3" target="_self">is_a_coproduct_wrt</a> <font color="Maroon">F</font> implies <font color="Maroon">a</font> is <a href="cat_1.html#V7">initial</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:93_1"/><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>
 ;<br/>

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

<b>consider </b><font color="Maroon">F'</font> being    <a href="cat_3.html#M2" target="_self">Injections_family</a> of <font color="Maroon">b</font>, <a href="xboole_0.html#K1">{}</a> ;<br/>
<a NAME="E3:93_1"/>
(  <a href="cat_3.html#K3" target="_self">doms</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="cat_3.html#K3" target="_self">doms</a> <font color="Maroon">F'</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="partfun1.html#T57">PARTFUN1:57</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="E5:93_1"/><i><font color="Green">E42</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="E6:93_1"/><i><font color="Green">E43</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>  <a href="xboole_0.html#K1">{}</a>  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="E7:93_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><a class="ref" href="cat_3.html#T9" target="_self">Th9</a></i>;<br/>

<b>reconsider </b><font color="Maroon">f</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_3.html#T9" target="_self">Th9</a>, <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>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/>

<a NAME="E9:93_1"/>
( <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> &amp; ( 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>  <a href="xboole_0.html#K1">{}</a>  holds <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="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> ) )
 
<b>by </b><i><a class="ref" href="cat_3.html#T9" target="_self">Th9</a>, <a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E10:93_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="cat_3.html#T10" target="_self">Th10</a></i>;<br/>


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

<b>assume </b><a NAME="E2:93"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">a</font> is <a href="cat_1.html#V7">initial</a>
 ;<br/>

<b>thus </b><a NAME="E3:93"/>
<font color="Maroon">F</font> is    <a href="cat_3.html#M2" target="_self">Injections_family</a> of <font color="Maroon">a</font>, <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_3.html#D18" target="_self">CAT_3:def 18</a><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_3.html#M2" target="_self">Injections_family</a> of <font color="Maroon">b</font>, <a href="xboole_0.html#K1">{}</a> ;<br/>



<b>assume </b><a NAME="E4:93"/>
 <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">h</font> being    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font><b> such that </b><br/><a NAME="E6:93"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">g</font> being   <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font> holds  <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font>
 <b>by </b><i>, <a class="ref" href="cat_1.html#D16">CAT_1:def 16</a></i>;<br/>
<b>take </b>
<font color="Maroon">h</font>
;<br/>

<a NAME="E7:93"/>
 <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>, <a class="ref" href="cat_1.html#D16">CAT_1:def 16</a></i>;<br/>
<b>hence </b><a NAME="E8:93"/>
<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>by </b><i><a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>

<b>let </b><font color="Maroon">k</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E9:93"/>
<font color="Maroon">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>
 ;<br/>

<a NAME="E10:93"/><b>then </b>
<font color="Maroon">k</font> is    <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>hence </b><a NAME="E11:93"/>
( ( 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>  <a href="xboole_0.html#K1">{}</a>  holds <br/><font color="Maroon">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="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="cat_3.html#T9" target="_self">Th9</a></i>;<br/>


</div>
