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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="cat_5.html#V3">Categorial</a> <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>





<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K12">Obj</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="mcart_1.html#K16">pr2</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>;<br/>
<a NAME="E1:55"/><i><font color="Green">E26</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K12">Obj</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="mcart_1.html#K16">pr2</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K12">Obj</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K12">Obj</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="mcart_1.html#K16">pr2</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a>  <a href="mcart_1.html#K16">pr2</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E2:55"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:55"/><b>then </b><i><font color="Green">E27</font></i>: 
 <a href="cat_1.html#K12">Obj</a> <font color="Maroon">c<sub>3</sub></font> is    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:55"/><i><font color="Green">E28</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="mcart_1.html#K16">pr2</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="mcart_1.html#D13">MCART_1:def 13</a></i>;<br/>
<a NAME="E5:55"/><b>then </b>
 <a href="mcart_1.html#K16">pr2</a> <font color="Maroon">c<sub>3</sub></font> is    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K12">Obj</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="mcart_1.html#K16">pr2</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> as    <a href="index_1.html#M1" target="_self">ManySortedSet</a> of the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="index_1.html#E3:55"><i><font color="Green">E27</font></i></a>, <a class="ref" href="index_1.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E7:55"/>
<font color="Maroon">c<sub>5</sub></font> <a href="index_1.html#K8" target="_self">`1</a>  is <a href="index_1.html#V1" target="_self">Category-yielding</a>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<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#V2" target="_self">Category-yielding_on_first</a>  <a href="index_1.html#M1" target="_self">ManySortedSet</a> of the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="index_1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E9:55"/>
 <a href="mcart_1.html#K16">pr2</a> <font color="Maroon">c<sub>3</sub></font> is <a href="funcop_1.html#V1">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">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funcop_1.html#D6">FUNCOP_1:def 6</a><br/>

<b>assume </b><a NAME="E1:55_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="mcart_1.html#K16">pr2</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="index_1.html#E4:55"><i><font color="Green">E28</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E4:55_1"/>
<span class="p1">(<span class="default"><a href="mcart_1.html#K16">pr2</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="index_1.html#K1" target="_self">`2</a> 
 
<b>by </b><i><a class="txt" href="index_1.html#E4:55"><i><font color="Green">E28</font></i></a>, <a class="ref" href="mcart_1.html#D13">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">pr2</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> is    <a href="hidden.html#M1">set</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="mcart_1.html#K16">pr2</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="pboole.html#NM2">ManySortedFunction</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="index_1.html#E4:55"><i><font color="Green">E28</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E11:55"/>
<font color="Maroon">c<sub>7</sub></font> is    <a href="index_1.html#M2" target="_self">ManySortedFunctor</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</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 class="p1">(<span class="default"><font color="Maroon">c<sub>6</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>
 
<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">c<sub>8</sub></font> be   <a href="cat_1.html#NM2">Morphism</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#D7" target="_self">INDEX_1:def 7</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E2:55_2"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="cat_5.html#K8">`11</a>  <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="cat_5.html#K9">`12</a>  <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="cat_5.html#T13">CAT_5:13</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>9</sub></font> <a href="cat_5.html#K8">`11</a> ,<font color="Maroon">c<sub>9</sub></font> <a href="cat_5.html#K9">`12</a> <b> such that </b><br/><a NAME="E4:55_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="cat_5.html#K8">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="cat_5.html#K9">`12</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="cat_5.html#D6">CAT_5:def 6</a></i>;<br/>
<i><font color="Green">E31</font></i>: <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> = 
<font color="Maroon">c<sub>9</sub></font> <a href="index_1.html#K1" target="_self">`2</a> 
<b>by </b><i><a class="txt" href="index_1.html#E4:55"><i><font color="Green">E28</font></i></a>, <a class="ref" href="mcart_1.html#D13">MCART_1:def 13</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="index_1.html#E4:55_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>
<i><font color="Green">E32</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K12">Obj</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> = 
<span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
 <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T105">CAT_1:105</a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K12">Obj</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> = 
<span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T105">CAT_1:105</a></i>
;<br/>
<b>hence </b><a NAME="E8:55_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</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>6</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="Maroon">c<sub>8</sub></font>,<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#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="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="index_1.html#E1:55"><i><font color="Green">E26</font></i></a>, <a class="txt" href="index_1.html#E2:55_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="index_1.html#E5:55_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="index_1.html#E6:55_2"><i><font color="Green">E32</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> 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> <b>by </b><i><a class="txt" href="index_1.html#E1:55"><i><font color="Green">E26</font></i></a>, <a class="ref" href="index_1.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E13:55"/><i><font color="Green">E29</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><i><font color="Green">E30</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">c<sub>9</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">(<span class="default"><a href="cat_1.html#K12">Obj</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> as   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>8</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"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="index_1.html#K1" target="_self">`2</a> 
<b>by </b><i><a class="txt" href="index_1.html#E1:55"><i><font color="Green">E26</font></i></a>, <a class="txt" href="index_1.html#E13:55"><i><font color="Green">E29</font></i></a>, <a class="ref" href="mcart_1.html#D13">MCART_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="index_1.html#K1" target="_self">`2</a> 
<b>by </b><i><a class="ref" href="cat_1.html#T104">CAT_1:104</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>9</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>9</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="cat_1.html#K15">id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>9</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="txt" href="index_1.html#E1:55"><i><font color="Green">E26</font></i></a>, <a class="ref" href="cat_5.html#D6">CAT_5:def 6</a></i>
<br/>.= 
 <a href="cat_1.html#K15">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">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">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</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="E1:55_4"/><i><font color="Green">E31</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><b>set </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font>;<br/><a NAME="E2:55_4"/><i><font color="Green">E32</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="cat_5.html#K8">`11</a>  &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="cat_5.html#K9">`12</a>  &amp;  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="cat_5.html#K8">`11</a>  &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="cat_5.html#K9">`12</a>  )
 <b>by </b><i><a class="ref" href="cat_5.html#T13">CAT_5:13</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="cat_1.html#M2">Functor</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="cat_5.html#K8">`11</a> ,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="cat_5.html#K9">`12</a> <b> such that </b><br/><a NAME="E4:55_4"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="cat_5.html#K8">`11</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="cat_5.html#K9">`12</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="cat_5.html#D6">CAT_5:def 6</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="cat_1.html#M2">Functor</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="cat_5.html#K8">`11</a> ,<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="cat_5.html#K9">`12</a> <b> such that </b><br/><a NAME="E6:55_4"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="cat_5.html#K8">`11</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="cat_5.html#K9">`12</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="index_1.html#E2:55_4"><i><font color="Green">E32</font></i></a>, <a class="ref" href="cat_5.html#D6">CAT_5:def 6</a></i>;<br/><i><font color="Green">E35</font></i>:  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> = 
<font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T109">CAT_1:109</a></i>
<br/>.= 
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="index_1.html#E1:55_4"><i><font color="Green">E31</font></i></a>, <a class="ref" href="cat_1.html#T109">CAT_1:109</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>8</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>10</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="index_1.html#K1" target="_self">`2</a> 
<b>by </b><i><a class="txt" href="index_1.html#E1:55"><i><font color="Green">E26</font></i></a>, <a class="txt" href="index_1.html#E13:55"><i><font color="Green">E29</font></i></a>, <a class="ref" href="mcart_1.html#D13">MCART_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="index_1.html#K1" target="_self">`2</a> 
<b>by </b><i><a class="txt" href="index_1.html#E1:55_4"><i><font color="Green">E31</font></i></a>, <a class="ref" href="cat_1.html#T99">CAT_1:99</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="cat_5.html#K8">`11</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="cat_5.html#K9">`12</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="txt" href="index_1.html#E2:55_4"><i><font color="Green">E32</font></i></a>, <a class="txt" href="index_1.html#E4:55_4"><i><font color="Green">E33</font></i></a>, <a class="txt" href="index_1.html#E6:55_4"><i><font color="Green">E34</font></i></a>, <a class="txt" href="index_1.html#E7:55_4"><i><font color="Green">E35</font></i></a>, <a class="ref" href="cat_5.html#D6">CAT_5:def 6</a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="index_1.html#K1" target="_self">`2</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="index_1.html#E4:55_4"><i><font color="Green">E33</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="index_1.html#K1" target="_self">`2</a> </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>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="index_1.html#K1" target="_self">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="index_1.html#E6:55_4"><i><font color="Green">E34</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</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>10</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>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="index_1.html#K1" target="_self">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="index_1.html#E1:55"><i><font color="Green">E26</font></i></a>, <a class="txt" href="index_1.html#E13:55"><i><font color="Green">E29</font></i></a>, <a class="ref" href="mcart_1.html#D13">MCART_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</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>10</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>8</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>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="index_1.html#E1:55"><i><font color="Green">E26</font></i></a>, <a class="txt" href="index_1.html#E13:55"><i><font color="Green">E29</font></i></a>, <a class="ref" href="mcart_1.html#D13">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">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K12">Obj</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="mcart_1.html#K16">pr2</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K4">]</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#E14:55"><i><font color="Green">E30</font></i></a>, <a class="ref" href="index_1.html#T6" target="_self">Th6</a></i>;<br/>


</div>
