<?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>let </b><font color="Maroon" title="c2">D</font> be  <a href="cat_5.html#V3" title="CAT_5:attr.3">Categorial</a> <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/><b>let </b><font color="Maroon" title="c3">F</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">D</font>;<br/>





<b>set </b><font color="Maroon" title="c4">I</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <font color="Maroon" title="c3">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c3">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<a NAME="E1:55"/><i><font color="Green" title="E26">A1</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <font color="Maroon" title="c3">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c3">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <font color="Maroon" title="c3">F</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <font color="Maroon" title="c3">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c3">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c3">F</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E2:55"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:55"/><b>then </b><i><font color="Green" title="E27">A2</font></i>: 
 <a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <font color="Maroon" title="c3">F</font> is    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:55"/><i><font color="Green" title="E28">A3</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">F</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="mcart_1.html#D13" title="MCART_1:def.13">MCART_1:def 13</a></i>;<br/>
<a NAME="E5:55"/><b>then </b>
 <a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c3">F</font> is    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">I</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <font color="Maroon" title="c3">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c3">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="index_1.html#M1" title="INDEX_1:mode.1">ManySortedSet</a> of 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> <b>by </b><i><a class="txt" href="index_1.html#E3:55"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="index_1.html#D4" target="_self" title="INDEX_1:def.4">Def4</a></i>;<br/>
<a NAME="E7:55"/>
<font color="Maroon" title="c5">I</font> <a href="index_1.html#K8" title="INDEX_1:func.8">`1</a>  is <a href="index_1.html#V1" title="INDEX_1:attr.1">Category-yielding</a>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">I</font> = <font color="Maroon" title="c5">I</font> as  <a href="index_1.html#V2" title="INDEX_1:attr.2">Category-yielding_on_first</a>  <a href="index_1.html#M1" title="INDEX_1:mode.1">ManySortedSet</a> of 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> <b>by </b><i><a class="ref" href="index_1.html#D5" target="_self" title="INDEX_1:def.5">Def5</a></i>;<br/>
<a NAME="E9:55"/>
 <a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c3">F</font> is <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funcop_1.html#D6" title="FUNCOP_1:def.6">FUNCOP_1:def 6</a><br/>

<b>assume </b><a NAME="E1:55_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c3">F</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">x</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="txt" href="index_1.html#E4:55"><i><font color="Green" title="E28">A3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">m</font> = <font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">D</font> ;<br/>
<a NAME="E4:55_1"/>
<span class="p1">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">m</font> <a href="index_1.html#K1" title="INDEX_1:func.1">`2</a> 
 
<b>by </b><i><a class="txt" href="index_1.html#E4:55"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="mcart_1.html#D13" title="MCART_1:def.13">MCART_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E5:55_1"/>
<span class="p1">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> is    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">FF</font> =  <a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c3">F</font> as   <a href="pboole.html#NM2" title="PBOOLE:NM.2">ManySortedFunction</a> of the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="txt" href="index_1.html#E4:55"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E11:55"/>
<font color="Maroon" title="c7">FF</font> is    <a href="index_1.html#M2" title="INDEX_1:mode.2">ManySortedFunctor</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="index_1.html#K10" title="INDEX_1:func.10">`1</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="index_1.html#K10" title="INDEX_1:func.10">`1</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">m</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="index_1.html#D7" target="_self" title="INDEX_1:def.7">INDEX_1:def 7</a><br/>

<b>reconsider </b><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">D</font> ;<br/>
<a NAME="E2:55_2"/><i><font color="Green" title="E29">A4</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="cat_5.html#K8" title="CAT_5:func.8">`11</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span> &amp; <font color="Maroon" title="c9">x</font> <a href="cat_5.html#K9" title="CAT_5:func.9">`12</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="cat_5.html#T13" title="CAT_5:th.13">CAT_5:13</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">f</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c9">x</font> <a href="cat_5.html#K8" title="CAT_5:func.8">`11</a> ,<font color="Maroon" title="c9">x</font> <a href="cat_5.html#K9" title="CAT_5:func.9">`12</a> <b> such that </b><br/><a NAME="E4:55_2"/><i><font color="Green" title="E30">A5</font></i>: 
<font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="cat_5.html#K8" title="CAT_5:func.8">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="cat_5.html#K9" title="CAT_5:func.9">`12</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c10">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="cat_5.html#D6" title="CAT_5:def.6">CAT_5:def 6</a></i>;<br/>
<i><font color="Green" title="E31">A6</font></i>: <font color="Maroon" title="c7">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font> = 
<font color="Maroon" title="c9">x</font> <a href="index_1.html#K1" title="INDEX_1:func.1">`2</a> 
<b>by </b><i><a class="txt" href="index_1.html#E4:55"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="mcart_1.html#D13" title="MCART_1:def.13">MCART_1:def 13</a></i>
<br/>.= 
<font color="Maroon" title="c10">f</font>
<b>by </b><i><a class="txt" href="index_1.html#E4:55_2"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/>
<i><font color="Green" title="E32">A7</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font> = 
<span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c8">m</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T105" title="CAT_1:th.105">CAT_1:105</a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font> = 
<span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c8">m</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T105" title="CAT_1:th.105">CAT_1:105</a></i>
;<br/>
<b>hence </b><a NAME="E8:55_2"/>
<font color="Maroon" title="c7">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font> is    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="index_1.html#K10" title="INDEX_1:func.10">`1</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <font color="Maroon" title="c1">C</font></span>)</span> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <font color="Maroon" title="c8">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="index_1.html#K10" title="INDEX_1:func.10">`1</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <font color="Maroon" title="c1">C</font></span>)</span> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <font color="Maroon" title="c8">m</font>
 <b>by </b><i><a class="txt" href="index_1.html#E1:55"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="index_1.html#E2:55_2"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="index_1.html#E5:55_2"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="index_1.html#E6:55_2"><i><font color="Green" title="E32">A7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c8">I</font> = <font color="Maroon" title="c6">I</font> as    <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> <b>by </b><i><a class="txt" href="index_1.html#E1:55"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="index_1.html#D8" target="_self" title="INDEX_1:def.8">Def8</a></i>;<br/>
<a NAME="E13:55"/><i><font color="Green" title="E29">A8</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<div><i><font color="Green" title="E30">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="55_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">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>reconsider </b><font color="Maroon" title="c10">i</font> = <span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">a</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">D</font> ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c8">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="c9">a</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c9">a</font></span>)</span></span>)</span> <a href="index_1.html#K1" title="INDEX_1:func.1">`2</a> 
<b>by </b><i><a class="txt" href="index_1.html#E1:55"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="index_1.html#E13:55"><i><font color="Green" title="E29">A8</font></i></a>, <a class="ref" href="mcart_1.html#D13" title="MCART_1:def.13">MCART_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c10">i</font></span>)</span> <a href="index_1.html#K1" title="INDEX_1:func.1">`2</a> 
<b>by </b><i><a class="ref" href="cat_1.html#T104" title="CAT_1:th.104">CAT_1:104</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">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="c9">a</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">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="c9">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">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="c9">a</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="index_1.html#E1:55"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="cat_5.html#D6" title="CAT_5:def.6">CAT_5:def 6</a></i>
<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="c8">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="c9">a</font></span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">m1</font>, <font color="Maroon" title="c10">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="E1:55_4"/><i><font color="Green" title="E31">A10</font></i>: 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">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="c9">m1</font>
 ;<br/><b>set </b><font color="Maroon" title="c11">h1</font> = <font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">m1</font>;<br/><b>set </b><font color="Maroon" title="c12">h2</font> = <font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m2</font>;<br/><a NAME="E2:55_4"/><i><font color="Green" title="E32">A11</font></i>: 
(  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m2</font></span>)</span> <a href="cat_5.html#K8" title="CAT_5:func.8">`11</a>  &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m2</font></span>)</span> <a href="cat_5.html#K9" title="CAT_5:func.9">`12</a>  &amp;  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">m1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">m1</font></span>)</span> <a href="cat_5.html#K8" title="CAT_5:func.8">`11</a>  &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">m1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">m1</font></span>)</span> <a href="cat_5.html#K9" title="CAT_5:func.9">`12</a>  )
 <b>by </b><i><a class="ref" href="cat_5.html#T13" title="CAT_5:th.13">CAT_5:13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">f2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m2</font></span>)</span> <a href="cat_5.html#K8" title="CAT_5:func.8">`11</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m2</font></span>)</span> <a href="cat_5.html#K9" title="CAT_5:func.9">`12</a> <b> such that </b><br/><a NAME="E4:55_4"/><i><font color="Green" title="E33">A12</font></i>: 
<font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m2</font></span>)</span> <a href="cat_5.html#K8" title="CAT_5:func.8">`11</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m2</font></span>)</span> <a href="cat_5.html#K9" title="CAT_5:func.9">`12</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c13">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="cat_5.html#D6" title="CAT_5:def.6">CAT_5:def 6</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">f1</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">m1</font></span>)</span> <a href="cat_5.html#K8" title="CAT_5:func.8">`11</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">m1</font></span>)</span> <a href="cat_5.html#K9" title="CAT_5:func.9">`12</a> <b> such that </b><br/><a NAME="E6:55_4"/><i><font color="Green" title="E34">A13</font></i>: 
<font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">m1</font></span>)</span> <a href="cat_5.html#K8" title="CAT_5:func.8">`11</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">m1</font></span>)</span> <a href="cat_5.html#K9" title="CAT_5:func.9">`12</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c14">f1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="index_1.html#E2:55_4"><i><font color="Green" title="E32">A11</font></i></a>, <a class="ref" href="cat_5.html#D6" title="CAT_5:def.6">CAT_5:def 6</a></i>;<br/><i><font color="Green" title="E35">A14</font></i>:  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m2</font></span>)</span> = 
<font color="Maroon" title="c3">F</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">m2</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T109" title="CAT_1:th.109">CAT_1:109</a></i>
<br/>.= 
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">m1</font></span>)</span>
<b>by </b><i><a class="txt" href="index_1.html#E1:55_4"><i><font color="Green" title="E31">A10</font></i></a>, <a class="ref" href="cat_1.html#T109" title="CAT_1:th.109">CAT_1:109</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c8">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="c10">m2</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c9">m1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m2</font> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <font color="Maroon" title="c9">m1</font></span>)</span></span>)</span> <a href="index_1.html#K1" title="INDEX_1:func.1">`2</a> 
<b>by </b><i><a class="txt" href="index_1.html#E1:55"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="index_1.html#E13:55"><i><font color="Green" title="E29">A8</font></i></a>, <a class="ref" href="mcart_1.html#D13" title="MCART_1:def.13">MCART_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m2</font></span>)</span> <a href="cat_1.html#K4" title="CAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">m1</font></span>)</span></span>)</span> <a href="index_1.html#K1" title="INDEX_1:func.1">`2</a> 
<b>by </b><i><a class="txt" href="index_1.html#E1:55_4"><i><font color="Green" title="E31">A10</font></i></a>, <a class="ref" href="cat_1.html#T99" title="CAT_1:th.99">CAT_1:99</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">m1</font></span>)</span> <a href="cat_5.html#K8" title="CAT_5:func.8">`11</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m2</font></span>)</span> <a href="cat_5.html#K9" title="CAT_5:func.9">`12</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c14">f1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="index_1.html#E2:55_4"><i><font color="Green" title="E32">A11</font></i></a>, <a class="txt" href="index_1.html#E4:55_4"><i><font color="Green" title="E33">A12</font></i></a>, <a class="txt" href="index_1.html#E6:55_4"><i><font color="Green" title="E34">A13</font></i></a>, <a class="txt" href="index_1.html#E7:55_4"><i><font color="Green" title="E35">A14</font></i></a>, <a class="ref" href="cat_5.html#D6" title="CAT_5:def.6">CAT_5:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c13">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c14">f1</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m2</font></span>)</span> <a href="index_1.html#K1" title="INDEX_1:func.1">`2</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c14">f1</font>
<b>by </b><i><a class="txt" href="index_1.html#E4:55_4"><i><font color="Green" title="E33">A12</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m2</font></span>)</span> <a href="index_1.html#K1" title="INDEX_1:func.1">`2</a> </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">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">m1</font></span>)</span> <a href="index_1.html#K1" title="INDEX_1:func.1">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="index_1.html#E6:55_4"><i><font color="Green" title="E34">A13</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">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="c10">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">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">m1</font></span>)</span> <a href="index_1.html#K1" title="INDEX_1:func.1">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="index_1.html#E1:55"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="index_1.html#E13:55"><i><font color="Green" title="E29">A8</font></i></a>, <a class="ref" href="mcart_1.html#D13" title="MCART_1:def.13">MCART_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">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="c10">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="c8">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="c9">m1</font></span>)</span>
<b>by </b><i><a class="txt" href="index_1.html#E1:55"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="index_1.html#E13:55"><i><font color="Green" title="E29">A8</font></i></a>, <a class="ref" href="mcart_1.html#D13" title="MCART_1:def.13">MCART_1:def 13</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E16:55"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <font color="Maroon" title="c3">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c3">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is   <a href="index_1.html#NM9" title="INDEX_1:NM.9">Indexing</a> of <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="index_1.html#E14:55"><i><font color="Green" title="E30">A9</font></i></a>, <a class="ref" href="index_1.html#T6" target="_self" title="INDEX_1:th.6">Th6</a></i>;<br/>


</div>
