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

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="cat_5.html#V2" title="CAT_5:attr.2">categorial</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Y</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st ( for <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font>, <font color="Olive" title="b4">C</font> being   <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b5">F</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">B</font><br/> for <font color="Olive" title="b6">G</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Olive" title="b3">B</font>,<font color="Olive" title="b4">C</font>  st <font color="Olive" title="b5">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> &amp; <font color="Olive" title="b6">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> holds <br/><font color="Olive" title="b6">G</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Olive" title="b5">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> ) &amp; ( for <font color="Olive" title="b2">A</font> being   <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds   <a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Olive" title="b2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> ) holds <br/> ex <font color="Olive" title="b2">C</font> being <a href="cat_1.html#V1" title="CAT_1:attr.1">strict</a> <a href="cat_5.html#V3" title="CAT_5:attr.3">Categorial</a> <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a> st <br/>( the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Olive" title="b2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; ( for <font color="Olive" title="b3">A</font>, <font color="Olive" title="b4">B</font> being   <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b5">F</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Olive" title="b3">A</font>,<font color="Olive" title="b4">B</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b3">A</font>,<font color="Olive" title="b4">B</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b5">F</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Olive" title="b2">C</font> iff <font color="Olive" title="b5">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> ) ) )</span><br/><b>let </b><font color="Maroon" title="c2">Y</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font>, <font color="Olive" title="b3">C</font> being   <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b4">F</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font><br/> for <font color="Olive" title="b5">G</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Olive" title="b2">B</font>,<font color="Olive" title="b3">C</font>  st <font color="Olive" title="b4">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Olive" title="b5">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> holds <br/><font color="Olive" title="b5">G</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Olive" title="b4">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> ) &amp; ( for <font color="Olive" title="b1">A</font> being   <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds   <a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> ) implies  ex <font color="Olive" title="b1">C</font> being <a href="cat_1.html#V1" title="CAT_1:attr.1">strict</a> <a href="cat_5.html#V3" title="CAT_5:attr.3">Categorial</a> <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a> st <br/>( the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; ( for <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font> being   <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b4">F</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">B</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">B</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b4">F</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">C</font> iff <font color="Olive" title="b4">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> ) ) ) )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:38"/><i><font color="Green" title="E24">A1</font></i>: 
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font>, <font color="Olive" title="b3">C</font> being   <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b4">F</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font><br/> for <font color="Olive" title="b5">G</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Olive" title="b2">B</font>,<font color="Olive" title="b3">C</font>  st <font color="Olive" title="b4">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Olive" title="b5">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> holds <br/><font color="Olive" title="b5">G</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Olive" title="b4">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 <b>and </b><br/><a NAME="E2:38"/><i><font color="Green" title="E25">A2</font></i>: 
for <font color="Olive" title="b1">A</font> being   <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds   <a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">C</font> being <a href="cat_1.html#V1" title="CAT_1:attr.1">strict</a> <a href="cat_5.html#V3" title="CAT_5:attr.3">Categorial</a> <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a> st <br/>( the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; ( for <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font> being   <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b4">F</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">B</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">B</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b4">F</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">C</font> iff <font color="Olive" title="b4">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> ) ) )</span><br/>

<b>set </b><font color="Maroon" title="c3">B</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> : <font color="Olive" title="b1">b</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> </span>}</span> ;<br/>
<b>consider </b><font color="Maroon" title="c4">a</font> being    <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>
<a NAME="E4:38"/>
 <a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="txt" href="cat_5.html#E2:38"><i><font color="Green" title="E25">A2</font></i></a></i>;<br/>
<a NAME="E5:38"/><b>then </b>
 <a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> : <font color="Olive" title="b1">b</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">B</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> : <font color="Olive" title="b1">b</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E7:38"/>
<font color="Maroon" title="c5">B</font> is <a href="fraenkel.html#V1" title="FRAENKEL:attr.1">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" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="fraenkel.html#D1" title="FRAENKEL:def.1">FRAENKEL:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> or <font color="Maroon" title="c6">x</font> is    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  )</span><br/>

<b>assume </b><a NAME="E1:38_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">x</font> is    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> </span><br/>

<a NAME="E2:38_1"/><b>then </b>
 ex <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> st <br/>( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; <font color="Olive" title="b1">b</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> )
 
;<br/>
<b>hence </b><a NAME="E3:38_1"/>
<font color="Maroon" title="c6">x</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">B</font> = <font color="Maroon" title="c5">B</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="fraenkel.html#V1" title="FRAENKEL:attr.1">functional</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">A</font> = <font color="Maroon" title="c1">X</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="cat_5.html#V2" title="CAT_5:attr.2">categorial</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c7">A</font>,   <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c7">A</font>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$3 is    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of $1,$2;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>,  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = $1 <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> $2;<br/>
<a NAME="E10:38"/><i><font color="Green" title="E26">A3</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c7">A</font><br/> for <font color="Olive" title="b4">f</font>, <font color="Olive" title="b5">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">B</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b4">f</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b5">g</font>] holds <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b5">g</font>,<font color="Olive" title="b4">f</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b5">g</font>,<font color="Olive" title="b4">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" title="c8">a</font>, <font color="Maroon" title="c9">b</font>, <font color="Maroon" title="c10">c</font> be    <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c7">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">B</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>,<font color="Olive" title="b1">f</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c10">c</font>,<font color="Olive" title="b2">g</font>] holds <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c10">c</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font>)] )</span><br/><b>let </b><font color="Maroon" title="c11">f</font>, <font color="Maroon" title="c12">g</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c11">f</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c12">g</font>] implies ( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c12">g</font>,<font color="Maroon" title="c11">f</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c10">c</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c12">g</font>,<font color="Maroon" title="c11">f</font>)] ) )</span><br/>



<b>assume </b><a NAME="E1:38_2"/><i><font color="Green" title="E27">A4</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c11">f</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c12">g</font>] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c12">g</font>,<font color="Maroon" title="c11">f</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c10">c</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c12">g</font>,<font color="Maroon" title="c11">f</font>)] )</span><br/>

<b>then reconsider </b><font color="Maroon" title="c13">f</font> = <font color="Maroon" title="c11">f</font> as    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c14">g</font> = <font color="Maroon" title="c12">g</font> as    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c10">c</font> <b>by </b><i><a class="txt" href="cat_5.html#E1:38_2"><i><font color="Green" title="E27">A4</font></i></a></i>;<br/>
<a NAME="E4:38_2"/>
( <font color="Maroon" title="c13">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> &amp; <font color="Maroon" title="c14">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> )
 
;<br/>
<a NAME="E5:38_2"/><b>then </b>
(  ex <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> st <br/>( <font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; <font color="Olive" title="b1">b</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ) &amp;  ex <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> st <br/>( <font color="Maroon" title="c14">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; <font color="Olive" title="b1">b</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ) )
 
;<br/>
<a NAME="E6:38_2"/><b>then </b>
<font color="Maroon" title="c14">g</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c13">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="txt" href="cat_5.html#E1:38"><i><font color="Green" title="E24">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:38_2"/>
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c12">g</font>,<font color="Maroon" title="c11">f</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c10">c</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c12">g</font>,<font color="Maroon" title="c11">f</font>)] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E11:38"/><i><font color="Green" title="E27">A5</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c7">A</font>  ex <font color="Olive" title="b2">f</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">B</font> st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">f</font>] &amp; ( for <font color="Olive" title="b3">b</font> being   <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c7">A</font><br/> for <font color="Olive" title="b4">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">B</font> holds <br/> ( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">g</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b4">g</font>,<font color="Olive" title="b2">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">g</font> ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">b</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b4">g</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">f</font>,<font color="Olive" title="b4">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">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" title="c8">a</font> be    <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c7">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">f</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">B</font> st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c8">a</font>,<font color="Olive" title="b1">f</font>] &amp; ( for <font color="Olive" title="b2">b</font> being   <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c7">A</font><br/> for <font color="Olive" title="b3">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">B</font> holds <br/> ( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">g</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b3">g</font>,<font color="Olive" title="b1">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">b</font>,<font color="Maroon" title="c8">a</font>,<font color="Olive" title="b3">g</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> ) ) ) )</span><br/>

<b>reconsider </b><font color="Maroon" title="c9">f</font> =  <a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c8">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> <b>by </b><i><a class="txt" href="cat_5.html#E2:38"><i><font color="Green" title="E25">A2</font></i></a></i>;<br/>
<a NAME="E2:38_3"/>
<font color="Maroon" title="c9">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">f</font> = <font color="Maroon" title="c9">f</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">B</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c10">f</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c10">f</font>] &amp; ( for <font color="Olive" title="b1">b</font> being   <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c7">A</font><br/> for <font color="Olive" title="b2">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">B</font> holds <br/> ( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">a</font>,<font color="Olive" title="b1">b</font>,<font color="Olive" title="b2">g</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">g</font>,<font color="Maroon" title="c10">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c8">a</font>,<font color="Olive" title="b2">g</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">f</font>,<font color="Olive" title="b2">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) ) ) )</span><br/>

<b>thus </b><a NAME="E4:38_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c10">f</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">b</font> being   <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c7">A</font><br/> for <font color="Olive" title="b2">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">B</font> holds <br/> ( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">a</font>,<font color="Olive" title="b1">b</font>,<font color="Olive" title="b2">g</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">g</font>,<font color="Maroon" title="c10">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c8">a</font>,<font color="Olive" title="b2">g</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">f</font>,<font color="Olive" title="b2">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) )</span><br/>

<b>let </b><font color="Maroon" title="c11">b</font> be    <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c7">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">B</font> holds <br/> ( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c11">b</font>,<font color="Olive" title="b1">g</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">g</font>,<font color="Maroon" title="c10">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c8">a</font>,<font color="Olive" title="b1">g</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">f</font>,<font color="Olive" title="b1">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> ) )</span><br/><b>let </b><font color="Maroon" title="c12">g</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c12">g</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c12">g</font>,<font color="Maroon" title="c10">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">g</font> ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c12">g</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c12">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">g</font> ) )</span><br/>



<b>thus </b><a NAME="E5:38_3"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c12">g</font>] implies <font color="Maroon" title="c12">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">g</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c12">g</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c12">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">g</font> )</span><br/>

<b>assume </b><a NAME="E6:38_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c12">g</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c12">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">g</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c13">G</font> = <font color="Maroon" title="c12">g</font> as    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c8">a</font> ;<br/>
<a NAME="E8:38_3"/>
<span class="p1">(<span class="default"><a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c13">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c13">G</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></i>;<br/>
<b>hence </b><a NAME="E9:38_3"/>
<font color="Maroon" title="c10">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c12">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E12:38"/><i><font color="Green" title="E28">A6</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font> being   <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c7">A</font><br/> for <font color="Olive" title="b5">f</font>, <font color="Olive" title="b6">g</font>, <font color="Olive" title="b7">h</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">B</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b5">f</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b6">g</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font>,<font color="Olive" title="b7">h</font>] holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b7">h</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b6">g</font>,<font color="Olive" title="b5">f</font>)) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b7">h</font>,<font color="Olive" title="b6">g</font>),<font color="Olive" title="b5">f</font>)
 
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">C</font> being  <a href="cat_1.html#V1" title="CAT_1:attr.1">strict</a> <a href="cat_5.html#V1" title="CAT_5:attr.1">with_triple-like_morphisms</a> <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a><b> such that </b><br/><a NAME="E14:38"/><i><font color="Green" title="E29">A7</font></i>: 
( the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c8">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c7">A</font><br/> for <font color="Olive" title="b3">f</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">B</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">f</font>] holds <br/><span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b3">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">C</font> ) &amp; ( for <font color="Olive" title="b1">m</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">C</font>  ex <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c7">A</font> ex <font color="Olive" title="b4">f</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">B</font> st <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b4">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">f</font>] ) ) &amp; ( for <font color="Olive" title="b1">m1</font>, <font color="Olive" title="b2">m2</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">C</font><br/> for <font color="Olive" title="b3">a1</font>, <font color="Olive" title="b4">a2</font>, <font color="Olive" title="b5">a3</font> being   <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c7">A</font><br/> for <font color="Olive" title="b6">f1</font>, <font color="Olive" title="b7">f2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">B</font>  st <font color="Olive" title="b1">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b3">a1</font>,<font color="Olive" title="b4">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b6">f1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Olive" title="b2">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b4">a2</font>,<font color="Olive" title="b5">a3</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b7">f2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> holds <br/><font color="Olive" title="b2">m2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b1">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b3">a1</font>,<font color="Olive" title="b5">a3</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b7">f2</font>,<font color="Olive" title="b6">f1</font>)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )
 <b>from </b><i><a class="ref" href="cat_5.html#S1" target="_self" title="CAT_5:sch.1">CAT_5:sch 1</a>(<a class="txt" href="cat_5.html#E10:38"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="cat_5.html#E11:38"><i><font color="Green" title="E27">A5</font></i></a>, <a class="txt" href="cat_5.html#E12:38"><i><font color="Green" title="E28">A6</font></i></a>);<br/></i>
<a NAME="E15:38"/>
<font color="Maroon" title="c8">C</font> is <a href="cat_5.html#V3" title="CAT_5:attr.3">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="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c8">C</font> is <a href="cat_5.html#V2" title="CAT_5:attr.2">categorial</a>
 <b>by </b><i><a class="txt" href="cat_5.html#E14:38"><i><font color="Green" title="E29">A7</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_5.html#D6" target="_self" title="CAT_5:def.6">CAT_5:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c8">C</font><br/> for <font color="Olive" title="b2">A</font> being  <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">A</font> holds <br/> <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">A</font>,<font color="Olive" title="b2">A</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Olive" title="b2">A</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) &amp; ( for <font color="Olive" title="b1">m</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">C</font><br/> for <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font> being  <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>  st <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Olive" title="b1">m</font> holds <br/> ex <font color="Olive" title="b4">F</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">B</font> st <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b4">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) &amp; ( for <font color="Olive" title="b1">m1</font>, <font color="Olive" title="b2">m2</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">C</font><br/> for <font color="Olive" title="b3">A</font>, <font color="Olive" title="b4">B</font>, <font color="Olive" title="b5">C</font> being  <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a><br/> for <font color="Olive" title="b6">F</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Olive" title="b3">A</font>,<font color="Olive" title="b4">B</font><br/> for <font color="Olive" title="b7">G</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Olive" title="b4">B</font>,<font color="Olive" title="b5">C</font>  st <font color="Olive" title="b1">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">A</font>,<font color="Olive" title="b4">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b6">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">B</font>,<font color="Olive" title="b5">C</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b7">G</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><font color="Olive" title="b2">m2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b1">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">A</font>,<font color="Olive" title="b5">C</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b7">G</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Olive" title="b6">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )</span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">m</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">C</font><br/> for <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font> being  <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>  st <font color="Olive" title="b2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Olive" title="b1">m</font> holds <br/> ex <font color="Olive" title="b4">F</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">B</font> st <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b4">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) &amp; ( for <font color="Olive" title="b1">m1</font>, <font color="Olive" title="b2">m2</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">C</font><br/> for <font color="Olive" title="b3">A</font>, <font color="Olive" title="b4">B</font>, <font color="Olive" title="b5">C</font> being  <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a><br/> for <font color="Olive" title="b6">F</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Olive" title="b3">A</font>,<font color="Olive" title="b4">B</font><br/> for <font color="Olive" title="b7">G</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Olive" title="b4">B</font>,<font color="Olive" title="b5">C</font>  st <font color="Olive" title="b1">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">A</font>,<font color="Olive" title="b4">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b6">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">B</font>,<font color="Olive" title="b5">C</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b7">G</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><font color="Olive" title="b2">m2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b1">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">A</font>,<font color="Olive" title="b5">C</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b7">G</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Olive" title="b6">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )</span>
<div class="add"><b>let </b><font color="Maroon" title="c9">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c8">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">D</font> being  <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>  st <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">D</font> holds <br/> <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">D</font>,<font color="Olive" title="b1">D</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Olive" title="b1">D</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/><b>let </b><font color="Maroon" title="c10">D</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">D</font> implies  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">D</font>,<font color="Maroon" title="c10">D</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c10">D</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/><b>assume </b><a NAME="E1:38_4_1"/><i><font color="Green" title="E30">A8</font></i>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">D</font>,<font color="Maroon" title="c10">D</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c10">D</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/><a NAME="E2:38_4_1"/><b>then </b>
 <a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c10">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="txt" href="cat_5.html#E2:38"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="cat_5.html#E14:38"><i><font color="Green" title="E29">A7</font></i></a></i>;<br/><a NAME="E3:38_4_1"/><b>then </b>
 <a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c10">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c11">f</font> =  <a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c10">D</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">B</font> ;<br/><b>reconsider </b><font color="Maroon" title="c12">x</font> = <font color="Maroon" title="c9">a</font> as    <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c7">A</font> <b>by </b><i><a class="txt" href="cat_5.html#E14:38"><i><font color="Green" title="E29">A7</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">F</font> = <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c12">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c11">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">C</font> <b>by </b><i><a class="txt" href="cat_5.html#E14:38"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="cat_5.html#E1:38_4_1"><i><font color="Green" title="E30">A8</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c14">b</font>, <font color="Maroon" title="c15">c</font> being    <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c7">A</font>, <font color="Maroon" title="c16">g</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">B</font><b> such that </b><br/><a NAME="E8:38_4_1"/><i><font color="Green" title="E31">A9</font></i>: 
(  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c15">c</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c16">g</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c15">c</font>,<font color="Maroon" title="c16">g</font>] )
 <b>by </b><i><a class="txt" href="cat_5.html#E14:38"><i><font color="Green" title="E29">A7</font></i></a></i>;<br/><a NAME="E9:38_4_1"/><i><font color="Green" title="E32">A10</font></i>: 
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="cat_5.html#K4" title="CAT_5:func.4">`11</a>  &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="cat_5.html#K5" title="CAT_5:func.5">`12</a>  &amp; <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="cat_5.html#K4" title="CAT_5:func.4">`11</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">b</font> &amp; <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="cat_5.html#K5" title="CAT_5:func.5">`12</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">c</font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E8:38_4_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="cat_5.html#T2" target="_self" title="CAT_5:th.2">Th2</a>, <a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></i>;<br/> <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c13">F</font> = 
<font color="Maroon" title="c13">F</font> <a href="cat_5.html#K5" title="CAT_5:func.5">`12</a> 
<b>by </b><i><a class="ref" href="cat_5.html#T2" target="_self" title="CAT_5:th.2">Th2</a></i>
<br/>.= 
<font color="Maroon" title="c12">x</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></i>
;<br/><b>then </b><font color="Maroon" title="c13">F</font> = 
<span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c13">F</font>
<b>by </b><i><a class="ref" href="cat_1.html#T46" title="CAT_1:th.46">CAT_1:46</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c15">c</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c10">D</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="cat_5.html#E14:38"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="cat_5.html#E8:38_4_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="txt" href="cat_5.html#E9:38_4_1"><i><font color="Green" title="E32">A10</font></i></a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c15">c</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c16">g</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="cat_5.html#E1:38_4_1"><i><font color="Green" title="E30">A8</font></i></a>, <a class="txt" href="cat_5.html#E8:38_4_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="txt" href="cat_5.html#E9:38_4_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></i>
;<br/><b>hence </b><a NAME="E12:38_4_1"/>
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">D</font>,<font color="Maroon" title="c10">D</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="cat_1.html#K15" title="CAT_1:func.15">id</a> <font color="Maroon" title="c10">D</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="cat_5.html#E1:38_4_1"><i><font color="Green" title="E30">A8</font></i></a>, <a class="txt" href="cat_5.html#E8:38_4_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="txt" href="cat_5.html#E9:38_4_1"><i><font color="Green" title="E32">A10</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">m1</font>, <font color="Olive" title="b2">m2</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">C</font><br/> for <font color="Olive" title="b3">A</font>, <font color="Olive" title="b4">B</font>, <font color="Olive" title="b5">C</font> being  <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a><br/> for <font color="Olive" title="b6">F</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Olive" title="b3">A</font>,<font color="Olive" title="b4">B</font><br/> for <font color="Olive" title="b7">G</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Olive" title="b4">B</font>,<font color="Olive" title="b5">C</font>  st <font color="Olive" title="b1">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">A</font>,<font color="Olive" title="b4">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b6">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">B</font>,<font color="Olive" title="b5">C</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b7">G</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><font color="Olive" title="b2">m2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b1">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">A</font>,<font color="Olive" title="b5">C</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b7">G</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Olive" title="b6">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>
<div class="add"><b>let </b><font color="Maroon" title="c9">m</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c9">m</font> &amp; <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c9">m</font> holds <br/> ex <font color="Olive" title="b3">F</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> st <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/><b>consider </b><font color="Maroon" title="c10">a</font>, <font color="Maroon" title="c11">b</font> being    <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c7">A</font>, <font color="Maroon" title="c12">f</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">B</font><b> such that </b><br/><a NAME="E2:38_4_2"/><i><font color="Green" title="E30">A11</font></i>: 
( <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c12">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c12">f</font>] )
 <b>by </b><i><a class="txt" href="cat_5.html#E14:38"><i><font color="Green" title="E29">A7</font></i></a></i>;<br/><a NAME="E3:38_4_2"/>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">m</font> <a href="cat_5.html#K4" title="CAT_5:func.4">`11</a>  &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">m</font> <a href="cat_5.html#K5" title="CAT_5:func.5">`12</a>  )
 <b>by </b><i><a class="ref" href="cat_5.html#T2" target="_self" title="CAT_5:th.2">Th2</a></i>;<br/><a NAME="E4:38_4_2"/><b>then </b>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b</font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E2:38_4_2"><i><font color="Green" title="E30">A11</font></i></a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></i>;<br/><b>hence </b><a NAME="E5:38_4_2"/>
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c9">m</font> &amp; <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c9">m</font> holds <br/> ex <font color="Olive" title="b3">F</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> st <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="cat_5.html#E2:38_4_2"><i><font color="Green" title="E30">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c9">m1</font>, <font color="Maroon" title="c10">m2</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font>, <font color="Olive" title="b3">C</font> being  <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a><br/> for <font color="Olive" title="b4">F</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font><br/> for <font color="Olive" title="b5">G</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Olive" title="b2">B</font>,<font color="Olive" title="b3">C</font>  st <font color="Maroon" title="c9">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b4">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c10">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">B</font>,<font color="Olive" title="b3">C</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b5">G</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><font color="Maroon" title="c10">m2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c9">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b3">C</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">G</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Olive" title="b4">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/>

<b>consider </b><font color="Maroon" title="c11">a1</font>, <font color="Maroon" title="c12">b1</font> being    <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c7">A</font>, <font color="Maroon" title="c13">f1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">B</font><b> such that </b><br/><a NAME="E5:38_4"/><i><font color="Green" title="E30">A12</font></i>: 
( <font color="Maroon" title="c9">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">a1</font>,<font color="Maroon" title="c12">b1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c13">f1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">a1</font>,<font color="Maroon" title="c12">b1</font>,<font color="Maroon" title="c13">f1</font>] )
 <b>by </b><i><a class="txt" href="cat_5.html#E14:38"><i><font color="Green" title="E29">A7</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">a2</font>, <font color="Maroon" title="c15">b2</font> being    <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c7">A</font>, <font color="Maroon" title="c16">f2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">B</font><b> such that </b><br/><a NAME="E7:38_4"/><i><font color="Green" title="E31">A13</font></i>: 
( <font color="Maroon" title="c10">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">a2</font>,<font color="Maroon" title="c15">b2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c16">f2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">a2</font>,<font color="Maroon" title="c15">b2</font>,<font color="Maroon" title="c16">f2</font>] )
 <b>by </b><i><a class="txt" href="cat_5.html#E14:38"><i><font color="Green" title="E29">A7</font></i></a></i>;<br/>
<b>let </b><font color="Maroon" title="c17">A</font>, <font color="Maroon" title="c18">B</font>, <font color="Maroon" title="c19">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c17">A</font>,<font color="Maroon" title="c18">B</font><br/> for <font color="Olive" title="b2">G</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c18">B</font>,<font color="Maroon" title="c19">C</font>  st <font color="Maroon" title="c9">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">A</font>,<font color="Maroon" title="c18">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b1">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c10">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">B</font>,<font color="Maroon" title="c19">C</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b2">G</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><font color="Maroon" title="c10">m2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c9">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">A</font>,<font color="Maroon" title="c19">C</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">G</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Olive" title="b1">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/><b>let </b><font color="Maroon" title="c20">F</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c17">A</font>,<font color="Maroon" title="c18">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">G</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c18">B</font>,<font color="Maroon" title="c19">C</font>  st <font color="Maroon" title="c9">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">A</font>,<font color="Maroon" title="c18">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c20">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c10">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">B</font>,<font color="Maroon" title="c19">C</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b1">G</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/><font color="Maroon" title="c10">m2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c9">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">A</font>,<font color="Maroon" title="c19">C</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">G</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c20">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/><b>let </b><font color="Maroon" title="c21">G</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c18">B</font>,<font color="Maroon" title="c19">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">A</font>,<font color="Maroon" title="c18">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c20">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c10">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">B</font>,<font color="Maroon" title="c19">C</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c21">G</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> implies <font color="Maroon" title="c10">m2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c9">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">A</font>,<font color="Maroon" title="c19">C</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c21">G</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c20">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>





<b>assume </b><a NAME="E8:38_4"/>
( <font color="Maroon" title="c9">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">A</font>,<font color="Maroon" title="c18">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c20">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c10">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">B</font>,<font color="Maroon" title="c19">C</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c21">G</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">m2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c9">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">A</font>,<font color="Maroon" title="c19">C</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c21">G</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c20">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/>

<a NAME="E9:38_4"/><b>then </b><i><font color="Green" title="E32">A14</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">A</font>,<font color="Maroon" title="c18">B</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">a1</font>,<font color="Maroon" title="c12">b1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">B</font>,<font color="Maroon" title="c19">C</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">a2</font>,<font color="Maroon" title="c15">b2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">F</font> &amp; <font color="Maroon" title="c16">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">G</font> )
 
<b>by </b><i><a class="txt" href="cat_5.html#E5:38_4"><i><font color="Green" title="E30">A12</font></i></a>, <a class="txt" href="cat_5.html#E7:38_4"><i><font color="Green" title="E31">A13</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E10:38_4"/><b>then </b>
( <font color="Maroon" title="c17">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">a1</font> &amp; <font color="Maroon" title="c18">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">b1</font> &amp; <font color="Maroon" title="c18">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a2</font> &amp; <font color="Maroon" title="c19">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">b2</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E11:38_4"/>
<font color="Maroon" title="c10">m2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c9">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">A</font>,<font color="Maroon" title="c19">C</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c21">G</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c20">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="cat_5.html#E14:38"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="cat_5.html#E5:38_4"><i><font color="Green" title="E30">A12</font></i></a>, <a class="txt" href="cat_5.html#E7:38_4"><i><font color="Green" title="E31">A13</font></i></a>, <a class="txt" href="cat_5.html#E9:38_4"><i><font color="Green" title="E32">A14</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c9">C</font> = <font color="Maroon" title="c8">C</font> as  <a href="cat_1.html#V1" title="CAT_1:attr.1">strict</a> <a href="cat_5.html#V3" title="CAT_5:attr.3">Categorial</a> <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a> ;<br/>
<b>take </b>
<font color="Maroon" title="c9">C</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; ( for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b3">F</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b3">F</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c9">C</font> iff <font color="Olive" title="b3">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> ) ) )</span><br/>

<b>thus </b><a NAME="E17:38"/>
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="cat_5.html#E14:38"><i><font color="Green" title="E29">A7</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b3">F</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b3">F</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c9">C</font> iff <font color="Olive" title="b3">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> )</span><br/>

<b>let </b><font color="Maroon" title="c10">A'</font>, <font color="Maroon" title="c11">B'</font> be    <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c10">A'</font>,<font color="Maroon" title="c11">B'</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">A'</font>,<font color="Maroon" title="c11">B'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b1">F</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c9">C</font> iff <font color="Olive" title="b1">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> )</span><br/><b>let </b><font color="Maroon" title="c12">F</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c10">A'</font>,<font color="Maroon" title="c11">B'</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">A'</font>,<font color="Maroon" title="c11">B'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c12">F</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c9">C</font> iff <font color="Maroon" title="c12">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> )</span><br/>



<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> implies <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">A'</font>,<font color="Maroon" title="c11">B'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c12">F</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c9">C</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:38_5"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">A'</font>,<font color="Maroon" title="c11">B'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c12">F</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c9">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font></span><br/><b>then reconsider </b><font color="Maroon" title="c13">m</font> = <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">A'</font>,<font color="Maroon" title="c11">B'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c12">F</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c9">C</font> ;<br/><b>consider </b><font color="Maroon" title="c14">a</font>, <font color="Maroon" title="c15">b</font> being    <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c7">A</font>, <font color="Maroon" title="c16">f</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">B</font><b> such that </b><br/><a NAME="E4:38_5"/><i><font color="Green" title="E30">A15</font></i>: 
( <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c16">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</font>,<font color="Maroon" title="c16">f</font>] )
 <b>by </b><i><a class="txt" href="cat_5.html#E14:38"><i><font color="Green" title="E29">A7</font></i></a></i>;<br/><a NAME="E5:38_5"/>
( <font color="Maroon" title="c13">m</font> <a href="cat_5.html#K11" title="CAT_5:func.11">`2</a>  <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c12">F</font> &amp; <font color="Maroon" title="c13">m</font> <a href="cat_5.html#K11" title="CAT_5:func.11">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">f</font> )
 <b>by </b><i><a class="txt" href="cat_5.html#E4:38_5"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/><a NAME="E6:38_5"/><b>then </b>
<font color="Maroon" title="c12">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font>
 ;<br/><a NAME="E7:38_5"/><b>then </b>
 ex <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">Y</font> st <br/>( <font color="Maroon" title="c12">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; <font color="Olive" title="b1">b</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> )
 ;<br/><b>hence </b><a NAME="E8:38_5"/>
<font color="Maroon" title="c12">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E19:38"/>
<font color="Maroon" title="c12">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">A'</font>,<font color="Maroon" title="c11">B'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c12">F</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c9">C</font></span><br/>

<a NAME="E20:38"/><b>then </b>
<font color="Maroon" title="c12">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font>
 
;<br/>
<b>hence </b><a NAME="E21:38"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">A'</font>,<font color="Maroon" title="c11">B'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c12">F</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c9">C</font>
 <b>by </b><i><a class="txt" href="cat_5.html#E14:38"><i><font color="Green" title="E29">A7</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
