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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="cat_5.html#V2" target="_self">categorial</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:38"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>4</sub></font> being   <a href="cat_1.html#M2">Functor</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font><br/>for <font color="Olive">b<sub>5</sub></font> being   <a href="cat_1.html#M2">Functor</a> of <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>5</sub></font> <a href="cat_1.html#K14">*</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:38"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds  <a href="cat_1.html#K15">id</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> is   <a href="funct_1.html#NM1">Function</a> </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E4:38"/>
 <a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="cat_5.html#E2:38"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E5:38"/><b>then </b>
 <a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> is   <a href="funct_1.html#NM1">Function</a> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> is   <a href="funct_1.html#NM1">Function</a> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E7:38"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="fraenkel.html#V1">functional</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="fraenkel.html#D1">FRAENKEL:def 1</a><br/>

<b>assume </b><a NAME="E1:38_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:38_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is   <a href="funct_1.html#NM1">Function</a> )
 
;<br/>
<b>hence </b><a NAME="E3:38_1"/>
<font color="Maroon">c<sub>6</sub></font> is   <a href="funct_1.html#NM1">Function</a>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="fraenkel.html#V1">functional</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="cat_5.html#V2" target="_self">categorial</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>7</sub></font>,   <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>7</sub></font>,   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>3</sub></font> is    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="funct_1.html#NM1">Function</a>,  <a href="funct_1.html#NM1">Function</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">a<sub>1</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<a NAME="E10:38"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>7</sub></font><br/>for <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>5</sub></font>] holds <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>4</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>4</sub></font>)] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>7</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font>;<br/>



<b>assume </b><a NAME="E1:38_2"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>12</sub></font>] )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font> as    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <b>by </b><i><a class="txt" href="cat_5.html#E1:38_2"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E4:38_2"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 
;<br/>
<a NAME="E5:38_2"/><b>then </b>
( ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is   <a href="funct_1.html#NM1">Function</a> ) &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is   <a href="funct_1.html#NM1">Function</a> ) )
 
;<br/>
<a NAME="E6:38_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="cat_5.html#E1:38"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:38_2"/>
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>11</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>11</sub></font>)] )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E11:38"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>7</sub></font>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>7</sub></font><br/>for <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>2</sub></font>) <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>4</sub></font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>) <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> ) ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>7</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> =  <a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>8</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="cat_5.html#E2:38"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E2:38_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/>

<b>thus </b><a NAME="E4:38_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font>]
 ;<br/>

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>7</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font>;<br/>



<b>thus </b><a NAME="E5:38_3"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>] implies <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>;<br/>

<b>assume </b><a NAME="E6:38_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>12</sub></font>]
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font> ;<br/>
<a NAME="E8:38_3"/>
<span class="p1">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>;<br/>
<b>hence </b><a NAME="E9:38_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E12:38"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>7</sub></font><br/>for <font color="Olive">b<sub>5</sub></font>, <font color="Olive">b<sub>6</sub></font>, <font color="Olive">b<sub>7</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>5</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>6</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>7</sub></font>] holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>7</sub></font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>6</sub></font>,<font color="Olive">b<sub>5</sub></font>)) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>7</sub></font>,<font color="Olive">b<sub>6</sub></font>),<font color="Olive">b<sub>5</sub></font>)
 
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being  <a href="cat_1.html#V1">strict</a> <a href="cat_5.html#V1" target="_self">with_triple-like_morphisms</a> <a href="cat_1.html#NM4">Category</a><b> such that </b><br/><a NAME="E14:38"/><i><font color="Green">E29</font></i>: 
( the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>7</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>] holds <br/><span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">b<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span> is   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>8</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>8</sub></font>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>7</sub></font>ex <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">b<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>] ) ) &amp; ( 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>8</sub></font><br/>for <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_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>7</sub></font><br/>for <font color="Olive">b<sub>6</sub></font>, <font color="Olive">b<sub>7</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">b<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">b<sub>7</sub></font></span><a href="domain_1.html#K1">]</a></span> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="cat_1.html#K4">*</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="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>7</sub></font>,<font color="Olive">b<sub>6</sub></font>)</span><a href="tarski.html#K4">]</a></span> ) )
 <b>from </b><i><a class="ref" href="cat_5.html#S1" target="_self">CAT_5:sch 1</a>(<a class="txt" href="cat_5.html#E10:38"><i><font color="Green">E26</font></i></a>, <a class="txt" href="cat_5.html#E11:38"><i><font color="Green">E27</font></i></a>, <a class="txt" href="cat_5.html#E12:38"><i><font color="Green">E28</font></i></a>);<br/></i>
<a NAME="E15:38"/>
<font color="Maroon">c<sub>8</sub></font> is <a href="cat_5.html#V3" target="_self">Categorial</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:38_4"/>
the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>8</sub></font> is <a href="cat_5.html#V2" target="_self">categorial</a>
 <b>by </b><i><a class="txt" href="cat_5.html#E14:38"><i><font color="Green">E29</font></i></a></i>; <i><font color="Red">:: according to </font></i><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>9</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>8</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>assume </b><a NAME="E1:38_4_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><a NAME="E2:38_4_1"/><b>then </b>
 <a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="cat_5.html#E2:38"><i><font color="Green">E25</font></i></a>, <a class="txt" href="cat_5.html#E14:38"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E3:38_4_1"/><b>then </b>
 <a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> =  <a href="cat_1.html#K15">id</a> <font color="Maroon">c<sub>10</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> ;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="cat_5.html#E14:38"><i><font color="Green">E29</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K1">]</a></span> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>8</sub></font> <b>by </b><i><a class="txt" href="cat_5.html#E14:38"><i><font color="Green">E29</font></i></a>, <a class="txt" href="cat_5.html#E1:38_4_1"><i><font color="Green">E30</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> being    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>16</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E8:38_4_1"/><i><font color="Green">E31</font></i>: 
(  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">c<sub>16</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>] )
 <b>by </b><i><a class="txt" href="cat_5.html#E14:38"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E9:38_4_1"/><i><font color="Green">E32</font></i>: 
(  <a href="cat_1.html#K2">dom</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> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="cat_1.html#K3">cod</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> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="cat_1.html#K2">dom</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> <a href="hidden.html#R1">=</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> <a href="cat_5.html#K4" target="_self">`11</a>  &amp;  <a href="cat_1.html#K3">cod</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> <a href="hidden.html#R1">=</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> <a href="cat_5.html#K5" target="_self">`12</a>  &amp; <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="cat_5.html#K4" target="_self">`11</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="cat_5.html#K5" target="_self">`12</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E8:38_4_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="cat_5.html#T2" target="_self">Th2</a>, <a class="ref" href="cat_1.html#T44">CAT_1:44</a>, <a class="ref" href="mcart_1.html#T89">MCART_1:89</a></i>;<br/> <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>13</sub></font> = 
<font color="Maroon">c<sub>13</sub></font> <a href="cat_5.html#K5" target="_self">`12</a> 
<b>by </b><i><a class="ref" href="cat_5.html#T2" target="_self">Th2</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="ref" href="mcart_1.html#T89">MCART_1:89</a></i>
;<br/><b>then </b><font color="Maroon">c<sub>13</sub></font> = 
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="cat_1.html#T46">CAT_1:46</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="cat_5.html#E14:38"><i><font color="Green">E29</font></i></a>, <a class="txt" href="cat_5.html#E8:38_4_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="cat_5.html#E9:38_4_1"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p3">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">c<sub>16</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="cat_5.html#E1:38_4_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="cat_5.html#E8:38_4_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="cat_5.html#E9:38_4_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_2.html#T23">FUNCT_2:23</a></i>
;<br/><b>hence </b><a NAME="E12:38_4_1"/>
 <a href="cat_1.html#K10">id</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"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>10</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>10</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="cat_5.html#E1:38_4_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="cat_5.html#E8:38_4_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="cat_5.html#E9:38_4_1"><i><font color="Green">E32</font></i></a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>8</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> being    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>12</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E2:38_4_2"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>] )
 <b>by </b><i><a class="txt" href="cat_5.html#E14:38"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E3:38_4_2"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="cat_5.html#K4" target="_self">`11</a>  &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="cat_5.html#K5" target="_self">`12</a>  )
 <b>by </b><i><a class="ref" href="cat_5.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E4:38_4_2"/><b>then </b>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E2:38_4_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="mcart_1.html#T89">MCART_1:89</a></i>;<br/><b>hence </b><a NAME="E5:38_4_2"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="cat_1.html#NM4">Category</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>9</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being   <a href="cat_1.html#M2">Functor</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> st <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="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span>,<font color="Olive">b<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="cat_5.html#E2:38_4_2"><i><font color="Green">E30</font></i></a></i>;<br/></div>
<b>end;</b></div>

<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>8</sub></font>;<br/>

<b>consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>13</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E5:38_4"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">c<sub>13</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font>] )
 <b>by </b><i><a class="txt" href="cat_5.html#E14:38"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> being    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>16</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E7:38_4"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">c<sub>16</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>] )
 <b>by </b><i><a class="txt" href="cat_5.html#E14:38"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">c<sub>20</sub></font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>21</sub></font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>;<br/>





<b>assume </b><a NAME="E8:38_4"/>
( <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>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">c<sub>20</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <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>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">c<sub>21</sub></font></span><a href="tarski.html#K4">]</a></span> )
 ;<br/>

<a NAME="E9:38_4"/><b>then </b><i><font color="Green">E32</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 
<b>by </b><i><a class="txt" href="cat_5.html#E5:38_4"><i><font color="Green">E30</font></i></a>, <a class="txt" href="cat_5.html#E7:38_4"><i><font color="Green">E31</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E10:38_4"/><b>then </b>
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E11:38_4"/>
<font color="Maroon">c<sub>10</sub></font> <a href="cat_1.html#K4">*</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"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="cat_1.html#K14">*</a> <font color="Maroon">c<sub>20</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="cat_5.html#E14:38"><i><font color="Green">E29</font></i></a>, <a class="txt" href="cat_5.html#E5:38_4"><i><font color="Green">E30</font></i></a>, <a class="txt" href="cat_5.html#E7:38_4"><i><font color="Green">E31</font></i></a>, <a class="txt" href="cat_5.html#E9:38_4"><i><font color="Green">E32</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as  <a href="cat_1.html#V1">strict</a> <a href="cat_5.html#V3" target="_self">Categorial</a> <a href="cat_1.html#NM4">Category</a> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/>

<b>thus </b><a NAME="E17:38"/>
the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="cat_5.html#E14:38"><i><font color="Green">E29</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:38_5"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K1">]</a></span> is   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>9</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K1">]</a></span> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>9</sub></font> ;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> being    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>16</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E4:38_5"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">c<sub>16</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>] )
 <b>by </b><i><a class="txt" href="cat_5.html#E14:38"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E5:38_5"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="cat_5.html#K11" target="_self">`2</a>  <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="cat_5.html#K11" target="_self">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E4:38_5"><i><font color="Green">E30</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/><a NAME="E6:38_5"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E7:38_5"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is   <a href="funct_1.html#NM1">Function</a> )
 ;<br/><b>hence </b><a NAME="E8:38_5"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E19:38"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E20:38"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 
;<br/>
<b>hence </b><a NAME="E21:38"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K1">]</a></span> is   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="cat_5.html#E14:38"><i><font color="Green">E29</font></i></a></i>;<br/>


</div>
