<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/>

<b>reconsider </b><font color="Maroon" title="c2">D</font> =  <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font> #) as   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a> <b>by </b><i><a class="ref" href="cat_5.html#T1" title="CAT_5:th.1">CAT_5:1</a></i>;<br/>
<a NAME="E2:39"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c2">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font> #)
 
;<br/>
<b>let </b><font color="Maroon" title="c3">I</font> be    <a href="index_1.html#M3" title="INDEX_1:mode.3">Indexing</a> of the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:39_1"/><i><font color="Green" title="E16">A2</font></i>: 
<font color="Maroon" title="c3">I</font> is   <a href="index_1.html#NM9" title="INDEX_1:NM.9">Indexing</a> of <font color="Maroon" title="c1">C</font>
 ;<br/><div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c4">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>;<br/><b>thus </b><a NAME="E1:39_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="index_1.html#K14" title="INDEX_1:func.14">`2</a> </span>)</span> <a href="index_1.html#K13" title="INDEX_1:func.13">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="index_1.html#K10" title="INDEX_1:func.10">`1</a> </span>)</span> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <font color="Maroon" title="c4">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="index_1.html#E2:39"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="index_1.html#E1:39_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="index_1.html#D10" target="_self" title="INDEX_1:def.10">Def10</a></i>;<br/></div>
<b>end;</b></div><b>let </b><font color="Maroon" title="c4">m1</font>, <font color="Maroon" title="c5">m2</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>;<br/><b>assume </b><a NAME="E3:39_1"/><i><font color="Green" title="E17">A3</font></i>: 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c5">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c4">m1</font>
 ;<br/><a NAME="E4:39_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="index_1.html#K14" title="INDEX_1:func.14">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">m2</font>,<font color="Maroon" title="c4">m1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="index_1.html#K14" title="INDEX_1:func.14">`2</a> </span>)</span> <a href="index_1.html#K13" title="INDEX_1:func.13">.</a> <font color="Maroon" title="c5">m2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="index_1.html#K14" title="INDEX_1:func.14">`2</a> </span>)</span> <a href="index_1.html#K13" title="INDEX_1:func.13">.</a> <font color="Maroon" title="c4">m1</font></span>)</span>
 <b>by </b><i><a class="txt" href="index_1.html#E2:39"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="index_1.html#E1:39_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="index_1.html#D10" target="_self" title="INDEX_1:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E5:39_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="index_1.html#K14" title="INDEX_1:func.14">`2</a> </span>)</span> <a href="index_1.html#K13" title="INDEX_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m2</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c4">m1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="index_1.html#K14" title="INDEX_1:func.14">`2</a> </span>)</span> <a href="index_1.html#K13" title="INDEX_1:func.13">.</a> <font color="Maroon" title="c5">m2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="index_1.html#K14" title="INDEX_1:func.14">`2</a> </span>)</span> <a href="index_1.html#K13" title="INDEX_1:func.13">.</a> <font color="Maroon" title="c4">m1</font></span>)</span>
 <b>by </b><i><a class="txt" href="index_1.html#E3:39_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="cat_1.html#T41" title="CAT_1:th.41">CAT_1:41</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:39"/><i><font color="Green" title="E16">A4</font></i>: 
( ( for <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="index_1.html#K14" title="INDEX_1:func.14">`2</a> </span>)</span> <a href="index_1.html#K13" title="INDEX_1:func.13">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="index_1.html#K10" title="INDEX_1:func.10">`1</a> </span>)</span> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <font color="Olive" title="b1">a</font></span>)</span> ) &amp; ( for <font color="Olive" title="b1">m1</font>, <font color="Olive" title="b2">m2</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>  st  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Olive" title="b2">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Olive" title="b1">m1</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="index_1.html#K14" title="INDEX_1:func.14">`2</a> </span>)</span> <a href="index_1.html#K13" title="INDEX_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">m2</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Olive" title="b1">m1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="index_1.html#K14" title="INDEX_1:func.14">`2</a> </span>)</span> <a href="index_1.html#K13" title="INDEX_1:func.13">.</a> <font color="Olive" title="b2">m2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="index_1.html#K14" title="INDEX_1:func.14">`2</a> </span>)</span> <a href="index_1.html#K13" title="INDEX_1:func.13">.</a> <font color="Olive" title="b1">m1</font></span>)</span> ) )
 ;<br/>

<b>thus </b><a NAME="E5:39"/>
 ex <font color="Olive" title="b1">D</font> being  <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a> st <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font> #)
 <b>by </b><i><a class="txt" href="index_1.html#E2:39"><i><font color="Green" title="E15">A1</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="index_1.html#D10" target="_self" title="INDEX_1:def.10">INDEX_1:def 10</a><br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c4">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="index_1.html#K14" title="INDEX_1:func.14">`2</a> </span>)</span> <a href="index_1.html#K13" title="INDEX_1:func.13">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">a</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="index_1.html#K14" title="INDEX_1:func.14">`2</a> </span>)</span> <a href="index_1.html#K13" title="INDEX_1:func.13">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">a</font></span>)</span>

<br/>.= 
 <a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="index_1.html#K10" title="INDEX_1:func.10">`1</a> </span>)</span> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <font color="Maroon" title="c4">a</font></span>)</span>
<b>by </b><i><a class="txt" href="index_1.html#E4:39"><i><font color="Green" title="E16">A4</font></i></a></i>
;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c4">m1</font>, <font color="Maroon" title="c5">m2</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>;<br/>

<b>assume </b><a NAME="E7:39"/>
the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">m1</font>
 ;<br/>

<a NAME="E8:39"/><b>then </b><i><font color="Green" title="E17">A5</font></i>: 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c5">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c4">m1</font>
 
;<br/>
<a NAME="E9:39"/><b>then </b><i><font color="Green" title="E18">A6</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="index_1.html#K14" title="INDEX_1:func.14">`2</a> </span>)</span> <a href="index_1.html#K13" title="INDEX_1:func.13">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m2</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c4">m1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="index_1.html#K14" title="INDEX_1:func.14">`2</a> </span>)</span> <a href="index_1.html#K13" title="INDEX_1:func.13">.</a> <font color="Maroon" title="c5">m2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="index_1.html#K14" title="INDEX_1:func.14">`2</a> </span>)</span> <a href="index_1.html#K13" title="INDEX_1:func.13">.</a> <font color="Maroon" title="c4">m1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="index_1.html#E4:39"><i><font color="Green" title="E16">A4</font></i></a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="index_1.html#K14" title="INDEX_1:func.14">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c5">m2</font>,<font color="Maroon" title="c4">m1</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="index_1.html#K14" title="INDEX_1:func.14">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">m2</font>,<font color="Maroon" title="c4">m1</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="index_1.html#K14" title="INDEX_1:func.14">`2</a> </span>)</span> <a href="index_1.html#K13" title="INDEX_1:func.13">.</a> <font color="Maroon" title="c5">m2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="index_1.html#K14" title="INDEX_1:func.14">`2</a> </span>)</span> <a href="index_1.html#K13" title="INDEX_1:func.13">.</a> <font color="Maroon" title="c4">m1</font></span>)</span>
<b>by </b><i><a class="txt" href="index_1.html#E8:39"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="cat_1.html#T41" title="CAT_1:th.41">CAT_1:41</a>, <a class="txt" href="index_1.html#E9:39"><i><font color="Green" title="E18">A6</font></i></a></i>
;<br/>


</div>
