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<div class="add">

<b>let </b><font color="Maroon">C</font> be   <a href="cat_1.html#NM4">Category</a>;<br/>

<a NAME="E1:40"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">C</font> <a href="oppcat_1.html#K2">opp</a>  <a href="hidden.html#R1">=</a>  <a href="cat_1.html#G1">CatStr</a>(# the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>,<span class="p1">(<span class="default"><a href="oppcat_1.html#K1">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span>,the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font> #)
 
<b>by </b><i><a class="ref" href="oppcat_1.html#D1">OPPCAT_1:def 1</a></i>;<br/>
<b>let </b><font color="Maroon">I</font> be    <a href="index_1.html#M3" target="_self">Indexing</a> of the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:40_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">I</font> is   <a href="index_1.html#NM10">coIndexing</a> of <font color="Maroon">C</font>
 ;<br/><div><b>hereby </b>
<div class="add"><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>thus </b><a NAME="E1:40_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K14" target="_self">`2</a> </span>)</span> <a href="index_1.html#K13" target="_self">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#R1">=</a>  <a href="cat_1.html#K15">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K10" target="_self">`1</a> </span>)</span> <a href="index_1.html#K2" target="_self">.</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i>, , </i>;<br/></div>
<b>end;</b></div><b>let </b><font color="Maroon">m1</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>, <font color="Maroon">m2</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>;<br/><b>assume </b><a NAME="E3:40_1"/><i><font color="Green">E28</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">m2</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">m1</font>
 ;<br/><a NAME="E4:40_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K14" target="_self">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="oppcat_1.html#K1">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">m1</font>,<font color="Maroon">m2</font></span><a href="cat_2.html#K13">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K14" target="_self">`2</a> </span>)</span> <a href="index_1.html#K13" target="_self">.</a> <font color="Maroon">m1</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K14" target="_self">`2</a> </span>)</span> <a href="index_1.html#K13" target="_self">.</a> <font color="Maroon">m2</font></span>)</span> &amp; <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">m2</font>,<font color="Maroon">m1</font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> )
 <b>by </b><i>, , , , <a class="ref" href="cat_1.html#T40">CAT_1:40</a></i>;<br/><a NAME="E5:40_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K14" target="_self">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">m2</font>,<font color="Maroon">m1</font></span><a href="cat_2.html#K13">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K14" target="_self">`2</a> </span>)</span> <a href="index_1.html#K13" target="_self">.</a> <font color="Maroon">m1</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K14" target="_self">`2</a> </span>)</span> <a href="index_1.html#K13" target="_self">.</a> <font color="Maroon">m2</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#D2">FUNCT_4:def 2</a></i>;<br/><b>hence </b><a NAME="E6:40_1"/>
<span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K14" target="_self">`2</a> </span>)</span> <a href="index_1.html#K13" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">m2</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">m1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K14" target="_self">`2</a> </span>)</span> <a href="index_1.html#K13" target="_self">.</a> <font color="Maroon">m1</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K14" target="_self">`2</a> </span>)</span> <a href="index_1.html#K13" target="_self">.</a> <font color="Maroon">m2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="cat_1.html#T41">CAT_1:41</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:40"/><i><font color="Green">E30</font></i>: 
( ( for <font color="Olive">a</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font> holds  <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K14" target="_self">`2</a> </span>)</span> <a href="index_1.html#K13" target="_self">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Olive">a</font></span>)</span> <a href="funct_2.html#R1">=</a>  <a href="cat_1.html#K15">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K10" target="_self">`1</a> </span>)</span> <a href="index_1.html#K2" target="_self">.</a> <font color="Olive">a</font></span>)</span> ) &amp; ( for <font color="Olive">m1</font>, <font color="Olive">m2</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>  st  <a href="cat_1.html#K2">dom</a> <font color="Olive">m2</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Olive">m1</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K14" target="_self">`2</a> </span>)</span> <a href="index_1.html#K13" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">m2</font> <a href="cat_1.html#K4">*</a> <font color="Olive">m1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K14" target="_self">`2</a> </span>)</span> <a href="index_1.html#K13" target="_self">.</a> <font color="Olive">m1</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K14" target="_self">`2</a> </span>)</span> <a href="index_1.html#K13" target="_self">.</a> <font color="Olive">m2</font></span>)</span> ) )
 ;<br/>

<b>thus </b><a NAME="E4:40"/>
 ex <font color="Olive">D</font> being  <a href="cat_1.html#NM4">Category</a> st <font color="Olive">D</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#G1">CatStr</a>(# the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font>,<span class="p1">(<span class="default"><a href="oppcat_1.html#K1">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span>,the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font> #)
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="index_1.html#D10" target="_self">INDEX_1:def 10</a><br/>

<div><b>hereby </b>
<div class="add"><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>thus </b><span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K14" target="_self">`2</a> </span>)</span> <a href="index_1.html#K13" target="_self">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K14" target="_self">`2</a> </span>)</span> <a href="index_1.html#K13" target="_self">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a</font></span>)</span>

<br/>.= 
 <a href="cat_1.html#K15">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K10" target="_self">`1</a> </span>)</span> <a href="index_1.html#K2" target="_self">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i/>
;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">m1</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>, <font color="Maroon">m2</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E6:40"/>
the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m2</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m1</font>
 ;<br/>

<a NAME="E7:40"/><b>then </b>
the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">C</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">m2</font>
 
;<br/>
<a NAME="E8:40"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">m2</font>
 
;<br/>
<a NAME="E9:40"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K14" target="_self">`2</a> </span>)</span> <a href="index_1.html#K13" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">m1</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">m2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K14" target="_self">`2</a> </span>)</span> <a href="index_1.html#K13" target="_self">.</a> <font color="Maroon">m2</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K14" target="_self">`2</a> </span>)</span> <a href="index_1.html#K13" target="_self">.</a> <font color="Maroon">m1</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E10:40"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K14" target="_self">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">m1</font>,<font color="Maroon">m2</font></span><a href="cat_2.html#K13">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K14" target="_self">`2</a> </span>)</span> <a href="index_1.html#K13" target="_self">.</a> <font color="Maroon">m2</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K14" target="_self">`2</a> </span>)</span> <a href="index_1.html#K13" target="_self">.</a> <font color="Maroon">m1</font></span>)</span> &amp; <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">m1</font>,<font color="Maroon">m2</font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> )
 
<b>by </b><i>, <a class="ref" href="cat_1.html#T40">CAT_1:40</a>, <a class="ref" href="cat_1.html#T41">CAT_1:41</a></i>;<br/>
<b>hence </b><a NAME="E11:40"/>
<span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K14" target="_self">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="oppcat_1.html#K1">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">m2</font>,<font color="Maroon">m1</font></span><a href="cat_2.html#K13">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K14" target="_self">`2</a> </span>)</span> <a href="index_1.html#K13" target="_self">.</a> <font color="Maroon">m2</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="index_1.html#K14" target="_self">`2</a> </span>)</span> <a href="index_1.html#K13" target="_self">.</a> <font color="Maroon">m1</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#D2">FUNCT_4:def 2</a></i>;<br/>


</div>
