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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">F</font> be   <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>;<br/>







<b>thus </b><a NAME="E1:85"/>
( <font color="Maroon">F</font> is    <a href="cat_3.html#M1" target="_self">Projections_family</a> of <font color="Maroon">c</font>,<font color="Maroon">I</font> implies <font color="Maroon">F</font> <a href="cat_3.html#K5" target="_self">opp</a>  is    <a href="cat_3.html#M2" target="_self">Injections_family</a> of <font color="Maroon">c</font> <a href="oppcat_1.html#K3">opp</a> ,<font color="Maroon">I</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:85_1"/><i><font color="Green">E38</font></i>: 
 <a href="cat_3.html#K3" target="_self">doms</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">c</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_3.html#D14" target="_self">CAT_3:def 14</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_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="E1:85_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><b>hence </b><span class="p1">(<span class="default"><a href="cat_3.html#K4" target="_self">cods</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="cat_3.html#K5" target="_self">opp</a> </span>)</span></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"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="cat_3.html#K5" target="_self">opp</a> </span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><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> <a href="oppcat_1.html#K5">opp</a> </span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<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> <a href="oppcat_1.html#K3">opp</a> 
<b>by </b><i><a class="ref" href="oppcat_1.html#T11">OPPCAT_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="cat_3.html#K1" target="_self">--&gt;</a> <font color="Maroon">c</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span> <a href="oppcat_1.html#K3">opp</a> 
<b>by </b><i><a class="ref" href="cat_3.html#T7" target="_self">Th7</a>, , </i>
<br/>.= 
<font color="Maroon">c</font> <a href="oppcat_1.html#K3">opp</a> 
<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> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="oppcat_1.html#K3">opp</a> </span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:85_1"/>
 <a href="cat_3.html#K4" target="_self">cods</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="cat_3.html#K5" target="_self">opp</a> </span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">I</font> <a href="cat_3.html#K1" target="_self">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="oppcat_1.html#K3">opp</a> </span>)</span>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_3.html#D17" target="_self">CAT_3:def 17</a><br/>


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

<b>assume </b><a NAME="E2:85"/><i><font color="Green">E43</font></i>: 
 <a href="cat_3.html#K4" target="_self">cods</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="cat_3.html#K5" target="_self">opp</a> </span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">I</font> <a href="cat_3.html#K1" target="_self">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="oppcat_1.html#K3">opp</a> </span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_3.html#D17" target="_self">CAT_3:def 17</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_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="E1:85_2"/><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/><b>hence </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/>
<br/>.= 
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><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> <a href="oppcat_1.html#K5">opp</a> </span>)</span>
<b>by </b><i><a class="ref" href="oppcat_1.html#T9">OPPCAT_1:9</a></i>
<br/>.= 
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="cat_3.html#K5" target="_self">opp</a> </span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font></span>)</span>
<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> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="oppcat_1.html#K3">opp</a> </span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="cat_3.html#T7" target="_self">Th7</a>, , </i>
<br/>.= 
<font color="Maroon">c</font> <a href="oppcat_1.html#K3">opp</a> 
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="oppcat_1.html#D2">OPPCAT_1:def 2</a></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">c</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:85"/>
 <a href="cat_3.html#K3" target="_self">doms</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">c</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_3.html#D14" target="_self">CAT_3:def 14</a><br/>


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