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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1"><a href="index_1.html#K12" target="_self">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="index_1.html#K12" target="_self">]</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1"><a href="index_1.html#K12" target="_self">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="index_1.html#K12" target="_self">]</a></span>;<br/>
<a NAME="E1:53"/><i><font color="Green">E25</font></i>: 
( <span class="p1"><a href="index_1.html#K12" target="_self">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="index_1.html#K12" target="_self">]</a></span> <a href="index_1.html#K10" target="_self">`1</a>  <a href="pboole.html#R6">=</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="index_1.html#K12" target="_self">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="index_1.html#K12" target="_self">]</a></span> <a href="index_1.html#K11" target="_self">`2</a>  <a href="pboole.html#R6">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<div><i><font color="Green">E26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2"><a href="index_1.html#K12" target="_self">[</a><span class="default"><span class="p3">(<span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p4">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="index_1.html#K12" target="_self">]</a></span> <a href="index_1.html#K11" target="_self">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
 <a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="index_1.html#E1:53"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
 <a href="cat_1.html#K15">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="index_1.html#K12" target="_self">[</a><span class="default"><span class="p4">(<span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p4">(<span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p5">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="index_1.html#K12" target="_self">]</a></span> <a href="index_1.html#K10" target="_self">`1</a> </span>)</span> <a href="index_1.html#K2" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="index_1.html#E1:53"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E3:53"/>
<span class="p1"><a href="index_1.html#K12" target="_self">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="index_1.html#K12" target="_self">]</a></span> is    <a href="index_1.html#M3" target="_self">Indexing</a> of the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:53_2"/>
<span class="p1"><a href="index_1.html#K12" target="_self">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="index_1.html#K12" target="_self">]</a></span> <a href="index_1.html#K11" target="_self">`2</a>  is    <a href="index_1.html#M2" target="_self">ManySortedFunctor</a> of <span class="p1">(<span class="default"><span class="p2"><a href="index_1.html#K12" target="_self">[</a><span class="default"><span class="p3">(<span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p4">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="index_1.html#K12" target="_self">]</a></span> <a href="index_1.html#K10" target="_self">`1</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><span class="p2"><a href="index_1.html#K12" target="_self">[</a><span class="default"><span class="p3">(<span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p4">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="index_1.html#K12" target="_self">]</a></span> <a href="index_1.html#K10" target="_self">`1</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="index_1.html#D8" target="_self">INDEX_1:def 8</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:53_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><span class="p2"><a href="index_1.html#K12" target="_self">[</a><span class="default"><span class="p3">(<span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p4">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="index_1.html#K12" target="_self">]</a></span> <a href="index_1.html#K11" target="_self">`2</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> is    <a href="cat_1.html#M2">Functor</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="index_1.html#K12" target="_self">[</a><span class="default"><span class="p4">(<span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p4">(<span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p5">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="index_1.html#K12" target="_self">]</a></span> <a href="index_1.html#K10" target="_self">`1</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="index_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="index_1.html#K12" target="_self">[</a><span class="default"><span class="p4">(<span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p4">(<span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p5">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="index_1.html#K12" target="_self">]</a></span> <a href="index_1.html#K10" target="_self">`1</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="index_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="index_1.html#E32"><i><font color="Green">Lemma23</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="index_1.html#D7" target="_self">INDEX_1:def 7</a><br/>


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


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1"><a href="index_1.html#K12" target="_self">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="index_1.html#K12" target="_self">]</a></span> as    <a href="index_1.html#M3" target="_self">Indexing</a> of the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:53_3"/>
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:53_3"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>7</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#R1">=</a>  <a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>7</sub></font> <a href="funct_2.html#R1">=</a>  <a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>6</sub></font> <a href="funct_2.html#R1">=</a>  <a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="index_1.html#E1:53"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E3:53_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>7</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>7</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:53"/>
<span class="p1"><a href="index_1.html#K12" target="_self">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="index_1.html#K12" target="_self">]</a></span> is   <a href="index_1.html#NM9">Indexing</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="index_1.html#E2:53"><i><font color="Green">E26</font></i></a>, <a class="ref" href="index_1.html#T6" target="_self">Th6</a></i>;<br/>

<a NAME="E7:53"/>
<font color="Maroon">c<sub>5</sub></font> is    <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<sub>1</sub></font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:53_4"/>
<font color="Maroon">c<sub>5</sub></font> <a href="index_1.html#K14" target="_self">`2</a>  is    <a href="index_1.html#M2" target="_self">ManySortedFunctor</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="index_1.html#K10" target="_self">`1</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="index_1.html#K10" target="_self">`1</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="index_1.html#D8" target="_self">INDEX_1:def 8</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_4_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:53_4_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">b<sub>1</sub></font> is    <a href="cat_1.html#M2">Functor</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="index_1.html#K10" target="_self">`1</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="index_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="index_1.html#K10" target="_self">`1</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="index_1.html#K2" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="index_1.html#E33"><i><font color="Green">Lemma24</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="index_1.html#D7" target="_self">INDEX_1:def 7</a><br/>


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


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <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<sub>1</sub></font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:53_5"/>
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><a NAME="E2:53_5"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>8</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_2.html#R1">=</a>  <a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>8</sub></font> <a href="funct_2.html#R1">=</a>  <a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>7</sub></font> <a href="funct_2.html#R1">=</a>  <a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="index_1.html#E1:53"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E3:53_5"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>8</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>7</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:53"/>
<span class="p1"><a href="index_1.html#K12" target="_self">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="index_1.html#K12" target="_self">]</a></span> is   <a href="index_1.html#NM10">coIndexing</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="index_1.html#E2:53"><i><font color="Green">E26</font></i></a>, <a class="ref" href="index_1.html#T7" target="_self">Th7</a></i>;<br/>


</div>
