<?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>set </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="cat_1.html#K8">1Cat</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>;<br/>
<b>thus </b><a NAME="E1:27"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p3">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span> holds <font color="Olive">b<sub>1</sub></font> is   <a href="cat_1.html#NM4">Category</a>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_5.html#D5" target="_self">CAT_5:def 5</a>,<a class="ref" href="cat_5.html#D6" target="_self">CAT_5:def 6</a><br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p3">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>assume </b><a NAME="E1:27_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:27_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:27_1"/>
 <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>4</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"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p3">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>assume </b><a NAME="E1:27_2"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><a NAME="E2:27_2"/><b>then </b><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>1</sub></font> as    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> ;<br/><b>take </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font>;<br/><b>thus </b><a NAME="E4:27_2"/>
<font color="Maroon">c<sub>4</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"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="cat_5.html#E2:27_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p3">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>;<br/>







<b>assume </b><a NAME="E4:27"/>
( <font color="Maroon">c<sub>4</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"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>5</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"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> )
 ;<br/>

<a NAME="E5:27"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span> <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"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <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"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:27"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#R1">=</a>  <a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#R1">=</a>  <a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E7:27"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#R1">=</a>  <a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#T23">FUNCT_2:23</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E8:27"/>
<font color="Maroon">c<sub>5</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>4</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"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div>
