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

<b>let </b><font color="Maroon">X</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">Y</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">E17</font></i>: 
for <font color="Olive">A</font>, <font color="Olive">B</font>, <font color="Olive">C</font> being   <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">X</font><br/> for <font color="Olive">F</font> being   <a href="cat_1.html#M2">Functor</a> of <font color="Olive">A</font>,<font color="Olive">B</font><br/> for <font color="Olive">G</font> being   <a href="cat_1.html#M2">Functor</a> of <font color="Olive">B</font>,<font color="Olive">C</font>  st <font color="Olive">F</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/><font color="Olive">G</font> <a href="cat_1.html#K14">*</a> <font color="Olive">F</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E2:38"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">A</font> being   <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">X</font> holds   <a href="cat_1.html#K15">id</a> <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<b>set </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : <font color="Olive">b</font> is   <a href="funct_1.html#NM1">Function</a> </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">X</font>;<br/>
<a NAME="E4:38"/>
 <a href="cat_1.html#K15">id</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:38"/><b>then </b>
 <a href="cat_1.html#K15">id</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : <font color="Olive">b</font> is   <a href="funct_1.html#NM1">Function</a> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> : <font color="Olive">b</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">B</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">x</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">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

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


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">B</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">A</font> = <font color="Maroon">X</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">A</font>,   <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">A</font>,   <a href="hidden.html#M1">set</a> ] <b>means </b><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">E41</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">A</font><br/> for <font color="Olive">f</font>, <font color="Olive">g</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">f</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b</font>,<font color="Olive">c</font>,<font color="Olive">g</font>] holds <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">g</font>,<font color="Olive">f</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">c</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Olive">g</font>,<font color="Olive">f</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">a</font> be    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">b</font> be    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">c</font> be    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>, <font color="Maroon">g</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>;<br/>



<b>assume </b><a NAME="E1:38_2"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">f</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">g</font>] )
 ;<br/>

<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font> ;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">b</font>,<font color="Maroon">c</font> <b>by </b><i/>;<br/>
<a NAME="E4:38_2"/>
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 
;<br/>
<a NAME="E5:38_2"/><b>then </b>
(  ex <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> st <br/>( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> is   <a href="funct_1.html#NM1">Function</a> ) &amp;  ex <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> st <br/>( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> is   <a href="funct_1.html#NM1">Function</a> ) )
 
;<br/>
<a NAME="E6:38_2"/><b>then </b>
<font color="Maroon">g</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="cat_5.html#T2" target="_self">Th2</a></i>;<br/>
<b>hence </b><a NAME="E7:38_2"/>
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">g</font>,<font color="Maroon">f</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">g</font>,<font color="Maroon">f</font>)] )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E11:38"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">a</font> being   <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">A</font>  ex <font color="Olive">f</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">a</font>,<font color="Olive">f</font>] &amp; ( for <font color="Olive">b</font> being   <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">A</font><br/> for <font color="Olive">g</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> holds <br/> ( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">g</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">g</font>,<font color="Olive">f</font>) <a href="hidden.html#R1">=</a> <font color="Olive">g</font> ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b</font>,<font color="Olive">a</font>,<font color="Olive">g</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">f</font>,<font color="Olive">g</font>) <a href="hidden.html#R1">=</a> <font color="Olive">g</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">a</font> be    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">A</font>;<br/>

<b>reconsider </b><font color="Maroon">f</font> =  <a href="cat_1.html#K15">id</a> <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<a NAME="E2:38_3"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> ;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

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

<b>let </b><font color="Maroon">b</font> be    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">g</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>;<br/>



<b>thus </b><a NAME="E5:38_3"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">g</font>] implies <font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</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">b</font>,<font color="Maroon">a</font>,<font color="Maroon">g</font>]
 ;<br/>

<b>then reconsider </b><font color="Maroon">G</font> = <font color="Maroon">g</font> as    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">b</font>,<font color="Maroon">a</font> ;<br/>
<a NAME="E8:38_3"/>
<span class="p1">(<span class="default"><a href="cat_1.html#K15">id</a> <font color="Maroon">a</font></span>)</span> <a href="cat_1.html#K14">*</a> <font color="Maroon">G</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">G</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">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E12:38"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">d</font> being   <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">A</font><br/> for <font color="Olive">f</font>, <font color="Olive">g</font>, <font color="Olive">h</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">f</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b</font>,<font color="Olive">c</font>,<font color="Olive">g</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">c</font>,<font color="Olive">d</font>,<font color="Olive">h</font>] holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">h</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Olive">g</font>,<font color="Olive">f</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">h</font>,<font color="Olive">g</font>),<font color="Olive">f</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</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">E49</font></i>: 
( the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">A</font><br/> for <font color="Olive">f</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">f</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">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">f</font></span><a href="domain_1.html#K1">]</a></span> is   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font> ) &amp; ( for <font color="Olive">m</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>  ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">A</font> ex <font color="Olive">f</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> st <br/>( <font color="Olive">m</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">a</font>,<font color="Olive">b</font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">f</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">f</font>] ) ) &amp; ( for <font color="Olive">m1</font>, <font color="Olive">m2</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font><br/> for <font color="Olive">a1</font>, <font color="Olive">a2</font>, <font color="Olive">a3</font> being   <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">A</font><br/> for <font color="Olive">f1</font>, <font color="Olive">f2</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>  st <font color="Olive">m1</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">a1</font>,<font color="Olive">a2</font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">f1</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Olive">m2</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">a2</font>,<font color="Olive">a3</font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">f2</font></span><a href="domain_1.html#K1">]</a></span> holds <br/><font color="Olive">m2</font> <a href="cat_1.html#K4">*</a> <font color="Olive">m1</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">a1</font>,<font color="Olive">a3</font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">H<sub>1</sub></font>(<font color="Olive">f2</font>,<font color="Olive">f1</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>(, , );<br/></i>
<a NAME="E15:38"/>
<font color="Maroon">C</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</font> is <a href="cat_5.html#V2" target="_self">categorial</a>
 <b>by </b><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">a</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">D</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>assume </b><a NAME="E1:38_4_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 ;<br/><a NAME="E2:38_4_1"/><b>then </b>
 <a href="cat_1.html#K15">id</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:38_4_1"/><b>then </b>
 <a href="cat_1.html#K15">id</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><b>then reconsider </b><font color="Maroon">f</font> =  <a href="cat_1.html#K15">id</a> <font color="Maroon">D</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font> ;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font> as    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">F</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">x</font>,<font color="Maroon">x</font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font> <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">b</font> being    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">c</font> being    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">g</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E8:38_4_1"/><i><font color="Green">E53</font></i>: 
(  <a href="cat_1.html#K10">id</a> <font color="Maroon">a</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">b</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">g</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">g</font>] )
 <b>by </b><i/>;<br/><a NAME="E9:38_4_1"/><i><font color="Green">E54</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">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</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">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</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">a</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">a</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">a</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">a</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">a</font></span>)</span> <a href="cat_5.html#K4" target="_self">`11</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a</font></span>)</span> <a href="cat_5.html#K5" target="_self">`12</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 <b>by </b><i>, , <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">F</font> = 
<font color="Maroon">F</font> <a href="cat_5.html#K5" target="_self">`12</a> 
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</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">F</font> = 
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">F</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">x</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">g</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">D</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , </i>
<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">x</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">g</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , , <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">a</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">D</font>,<font color="Maroon">D</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">D</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , </i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">m</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>;<br/><b>consider </b><font color="Maroon">a</font> being    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">b</font> being    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">f</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E2:38_4_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">m</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">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">f</font>] )
 <b>by </b><i/>;<br/><a NAME="E3:38_4_2"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="cat_5.html#K4" target="_self">`11</a>  &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="cat_5.html#K5" target="_self">`12</a>  )
 <b>by </b><i/>;<br/><a NAME="E4:38_4_2"/><b>then </b>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <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">A</font>, <font color="Olive">B</font> being  <a href="cat_1.html#NM4">Category</a>  st <font color="Olive">A</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">m</font> &amp; <font color="Olive">B</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">m</font> holds <br/> ex <font color="Olive">F</font> being   <a href="cat_1.html#M2">Functor</a> of <font color="Olive">A</font>,<font color="Olive">B</font> st <font color="Maroon">m</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">A</font>,<font color="Olive">B</font></span><a href="tarski.html#K4">]</a></span>,<font color="Olive">F</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">m1</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>, <font color="Maroon">m2</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>;<br/>

<b>consider </b><font color="Maroon">a1</font> being    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">b1</font> being    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">f1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E5:38_4"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">m1</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">a1</font>,<font color="Maroon">b1</font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">f1</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a1</font>,<font color="Maroon">b1</font>,<font color="Maroon">f1</font>] )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">a2</font> being    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">b2</font> being    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">f2</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E7:38_4"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">m2</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">a2</font>,<font color="Maroon">b2</font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">f2</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a2</font>,<font color="Maroon">b2</font>,<font color="Maroon">f2</font>] )
 <b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">A</font> be   <a href="cat_1.html#NM4">Category</a>, <font color="Maroon">B</font> be   <a href="cat_1.html#NM4">Category</a>, <font color="Maroon">C</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">F</font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/><b>let </b><font color="Maroon">G</font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font>;<br/>





<b>assume </b><a NAME="E8:38_4"/>
( <font color="Maroon">m1</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">A</font>,<font color="Maroon">B</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">F</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">m2</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">B</font>,<font color="Maroon">C</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">G</font></span><a href="tarski.html#K4">]</a></span> )
 ;<br/>

<a NAME="E9:38_4"/><b>then </b><i><font color="Green">E65</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</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">a1</font>,<font color="Maroon">b1</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">B</font>,<font color="Maroon">C</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">a2</font>,<font color="Maroon">b2</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> &amp; <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> )
 
<b>by </b><i>, , <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">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</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">m2</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">m1</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">A</font>,<font color="Maroon">C</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">C</font> = <font color="Maroon">C</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</font>
;<br/>

<b>thus </b><a NAME="E17:38"/>
the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">A'</font> be    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">B'</font> be    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">F</font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A'</font>,<font color="Maroon">B'</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">A'</font>,<font color="Maroon">B'</font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">F</font></span><a href="domain_1.html#K1">]</a></span> is   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>
 ;<br/><b>then reconsider </b><font color="Maroon">m</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">A'</font>,<font color="Maroon">B'</font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">F</font></span><a href="domain_1.html#K1">]</a></span> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font> ;<br/><b>consider </b><font color="Maroon">a</font> being    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">b</font> being    <a href="cat_5.html#M1" target="_self">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">f</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E4:38_5"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">m</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">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">f</font>] )
 <b>by </b><i/>;<br/><a NAME="E5:38_5"/>
( <font color="Maroon">m</font> <a href="cat_5.html#K11" target="_self">`2</a>  <a href="funct_2.html#R1">=</a> <font color="Maroon">F</font> &amp; <font color="Maroon">m</font> <a href="cat_5.html#K11" target="_self">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> )
 <b>by </b><i>, <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">F</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><a NAME="E7:38_5"/><b>then </b>
 ex <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> st <br/>( <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> is   <a href="funct_1.html#NM1">Function</a> )
 ;<br/><b>hence </b><a NAME="E8:38_5"/>
<font color="Maroon">F</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/></div>
<b>end;</b></div>

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

<a NAME="E20:38"/><b>then </b>
<font color="Maroon">F</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</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">A'</font>,<font color="Maroon">B'</font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">F</font></span><a href="domain_1.html#K1">]</a></span> is   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>
 <b>by </b><i/>;<br/>


</div>
