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

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be  <a href="cat_1.html#V1">strict</a> <a href="cat_1.html#NM4">Category</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:59_1_2"/><i><font color="Green">E47</font></i>: 
the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_2.html#K7">Funct</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:59_1_2"/><i><font color="Green">E48</font></i>: 
the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E3:59_1_2"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font> holds <br/>(  <a href="cat_1.html#K2">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp;  <a href="cat_1.html#K3">cod</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a>  )
 <b>and </b><br/><a NAME="E4:59_1_2"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font> st  <a href="cat_1.html#K2">dom</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E5:59_1_2"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font> st <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>3</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being   <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>ex <font color="Olive">b<sub>6</sub></font> being   <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>ex <font color="Olive">b<sub>7</sub></font> being   <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font> st <br/>( <font color="Olive">b<sub>1</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="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span>,<font color="Olive">b<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">b<sub>2</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="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span>,<font color="Olive">b<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="cat_2.html#K13">]</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="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive">b<sub>7</sub></font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Olive">b<sub>6</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 <b>and </b><br/><a NAME="E6:59_1_2"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>3</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being   <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> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="cat_1.html#K10">id</a> <font color="Olive">b<sub>1</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="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" target="_self">id</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E7:59_1_2"/><i><font color="Green">E53</font></i>: 
the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_2.html#K7">Funct</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E8:59_1_2"/><i><font color="Green">E54</font></i>: 
the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E9:59_1_2"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>4</sub></font> holds <br/>(  <a href="cat_1.html#K2">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp;  <a href="cat_1.html#K3">cod</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a>  )
 <b>and </b><br/><a NAME="E10:59_1_2"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>4</sub></font> st  <a href="cat_1.html#K2">dom</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E11:59_1_2"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>4</sub></font> st <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>4</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being   <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>ex <font color="Olive">b<sub>6</sub></font> being   <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>ex <font color="Olive">b<sub>7</sub></font> being   <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font> st <br/>( <font color="Olive">b<sub>1</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="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span>,<font color="Olive">b<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">b<sub>2</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="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span>,<font color="Olive">b<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="cat_2.html#K13">]</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="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive">b<sub>7</sub></font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Olive">b<sub>6</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 <b>and </b><br/><a NAME="E12:59_1_2"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>4</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being   <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> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="cat_1.html#K10">id</a> <font color="Olive">b<sub>1</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="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" target="_self">id</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:59_1_2_1"/>
the <a href="cat_1.html#U2">Morphisms</a> of <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>4</sub></font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E2:59_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="nattra_1.html#E8:59_1_2"><i><font color="Green">E54</font></i></a></i>;<br/><b>thus </b><a NAME="E2:59_1_2_1"/>
the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>4</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:59_1_2_1_1"/>
the <a href="cat_1.html#U2">Morphisms</a> of <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>4</sub></font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E2:59_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="nattra_1.html#E8:59_1_2"><i><font color="Green">E54</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_2.html#D8">FUNCT_2:def 8</a><br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="nattra_1.html#E2:59_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="nattra_1.html#E8:59_1_2"><i><font color="Green">E54</font></i></a></i>;<br/>
<b>thus </b>the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> = 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>5</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="txt" href="nattra_1.html#E3:59_1_2"><i><font color="Green">E49</font></i></a></i>
<br/>.= 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="nattra_1.html#E9:59_1_2"><i><font color="Green">E55</font></i></a></i>
<br/>.= 
the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>

;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:59_1_2_1"/>
the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>4</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:59_1_2_1_2"/>
the <a href="cat_1.html#U2">Morphisms</a> of <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>4</sub></font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E2:59_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="nattra_1.html#E8:59_1_2"><i><font color="Green">E54</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_2.html#D8">FUNCT_2:def 8</a><br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="nattra_1.html#E2:59_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="nattra_1.html#E8:59_1_2"><i><font color="Green">E54</font></i></a></i>;<br/>
<b>thus </b>the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> = 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="txt" href="nattra_1.html#E3:59_1_2"><i><font color="Green">E49</font></i></a></i>
<br/>.= 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="nattra_1.html#E9:59_1_2"><i><font color="Green">E55</font></i></a></i>
<br/>.= 
the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>

;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_1_3"><b>now </b></a><div class="add"><a NAME="E1:59_1_2_1_3"/><i><font color="Green">E59</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>6</sub></font> =  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>4</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i><a class="txt" href="nattra_1.html#E2:59_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="nattra_1.html#E8:59_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/><a NAME="E1:59_1_2_1_3_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="mcart_1.html#K2">`2</a> , <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="mcart_1.html#K1">`1</a>  as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="mcart_1.html#K2">`2</a> , <font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="mcart_1.html#K1">`1</a>  as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="nattra_1.html#E2:59_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="nattra_1.html#E8:59_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="nattra_1.html#E1:59_1_2_1_3_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><div><i><font color="Green">E61</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_1_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_1_2_1_3_1_1"/>
<span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being    <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>, <font color="Maroon">c<sub>15</sub></font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>16</sub></font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font><b> such that </b><br/><a NAME="E3:59_1_2_1_3_1_1"/><i><font color="Green">E62</font></i>: 
( <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"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <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"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>and </b><br/><a NAME="E4:59_1_2_1_3_1_1"/>
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="cat_2.html#K13">]</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>12</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="nattra_1.html#E5:59_1_2"><i><font color="Green">E51</font></i></a></i>;<br/> <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>11</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="txt" href="nattra_1.html#E9:59_1_2"><i><font color="Green">E55</font></i></a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="txt" href="nattra_1.html#E3:59_1_2_1_3_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="txt" href="nattra_1.html#E3:59_1_2_1_3_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="nattra_1.html#E9:59_1_2"><i><font color="Green">E55</font></i></a></i>
;<br/><b>hence </b><a NAME="E6:59_1_2_1_3_1_1"/>
<span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E10:59_1_2"><i><font color="Green">E56</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_1_3_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:59_1_2_1_3_1_2"/>
<span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being    <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>, <font color="Maroon">c<sub>15</sub></font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>16</sub></font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font><b> such that </b><br/><a NAME="E3:59_1_2_1_3_1_2"/><i><font color="Green">E62</font></i>: 
( <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"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>11</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>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>and </b><br/><a NAME="E4:59_1_2_1_3_1_2"/>
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="cat_2.html#K13">]</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>12</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="nattra_1.html#E11:59_1_2"><i><font color="Green">E57</font></i></a></i>;<br/> <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>9</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="txt" href="nattra_1.html#E3:59_1_2"><i><font color="Green">E49</font></i></a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="txt" href="nattra_1.html#E3:59_1_2_1_3_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="txt" href="nattra_1.html#E3:59_1_2_1_3_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="nattra_1.html#E3:59_1_2"><i><font color="Green">E49</font></i></a></i>
;<br/><b>hence </b><a NAME="E6:59_1_2_1_3_1_2"/>
<span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E4:59_1_2"><i><font color="Green">E50</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:59_1_2_1_3_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> iff <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="nattra_1.html#E4:59_1_2_1_3_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:59_1_2_1_3"/><i><font color="Green">E60</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="subset_1.html#T8">SUBSET_1:8</a></i>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E5:59_1_2_1_3"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="mcart_1.html#K2">`2</a> , <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="mcart_1.html#K1">`1</a>  as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="nattra_1.html#E1:59_1_2_1_3"><i><font color="Green">E59</font></i></a>, <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/><a NAME="E7:59_1_2_1_3"/><i><font color="Green">E62</font></i>: 
<span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E1:59_1_2_1_3"><i><font color="Green">E59</font></i></a>, <a class="txt" href="nattra_1.html#E5:59_1_2_1_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being    <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>, <font color="Maroon">c<sub>13</sub></font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>14</sub></font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font><b> such that </b><br/><a NAME="E9:59_1_2_1_3"/><i><font color="Green">E63</font></i>: 
( <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"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <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"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="cat_2.html#K13">]</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>10</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="nattra_1.html#E5:59_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="nattra_1.html#E5:59_1_2_1_3"><i><font color="Green">E61</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> being    <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>, <font color="Maroon">c<sub>18</sub></font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>19</sub></font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font><b> such that </b><br/><a NAME="E11:59_1_2_1_3"/><i><font color="Green">E64</font></i>: 
( <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"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">c<sub>18</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <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"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="cat_2.html#K13">]</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>15</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="nattra_1.html#E2:59_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="nattra_1.html#E8:59_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="nattra_1.html#E11:59_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="nattra_1.html#E4:59_1_2_1_3"><i><font color="Green">E60</font></i></a>, <a class="txt" href="nattra_1.html#E5:59_1_2_1_3"><i><font color="Green">E61</font></i></a>, <a class="txt" href="nattra_1.html#E7:59_1_2_1_3"><i><font color="Green">E62</font></i></a></i>;<br/><a NAME="E12:59_1_2_1_3"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</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>15</sub></font>,<font color="Maroon">c<sub>16</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>11</sub></font>,<font color="Maroon">c<sub>12</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>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="txt" href="nattra_1.html#E9:59_1_2_1_3"><i><font color="Green">E63</font></i></a>, <a class="txt" href="nattra_1.html#E11:59_1_2_1_3"><i><font color="Green">E64</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E13:59_1_2_1_3"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="nattra_1.html#E9:59_1_2_1_3"><i><font color="Green">E63</font></i></a>, <a class="txt" href="nattra_1.html#E11:59_1_2_1_3"><i><font color="Green">E64</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E14:59_1_2_1_3"/>
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E7:59_1_2_1_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="nattra_1.html#E9:59_1_2_1_3"><i><font color="Green">E63</font></i></a>, <a class="txt" href="nattra_1.html#E11:59_1_2_1_3"><i><font color="Green">E64</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:59_1_2_1"/>
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/><b>thus </b><a NAME="E6:59_1_2_1"/>
the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R1">=</a> the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>4</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_1_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:59_1_2_1_4"/>
the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E1:59_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="nattra_1.html#E7:59_1_2"><i><font color="Green">E53</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_2.html#D8">FUNCT_2:def 8</a><br/>

<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>3</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as    <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> <b>by </b><i><a class="txt" href="nattra_1.html#E1:59_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="cat_2.html#D3">CAT_2:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="nattra_1.html#E1:59_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="nattra_1.html#E7:59_1_2"><i><font color="Green">E53</font></i></a></i>;<br/>
<b>thus </b>the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> = 
 <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>5</sub></font>

<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"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" target="_self">id</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="nattra_1.html#E6:59_1_2"><i><font color="Green">E52</font></i></a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="nattra_1.html#E12:59_1_2"><i><font color="Green">E58</font></i></a></i>
<br/>.= 
the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>

;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E14:59_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E1:59_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="nattra_1.html#E7:59_1_2"><i><font color="Green">E53</font></i></a></i>;<br/>


</div>
