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

<b>let </b><font color="Maroon" title="c1">V</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> ;<br/><b>let </b><font color="Maroon" title="c2">W</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>;<br/>



<a NAME="E1:58"/><i><font color="Green" title="E35">A1</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c2">W</font></span>)</span><br/> for <font color="Olive" title="b3">a'</font>, <font color="Olive" title="b4">b'</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c1">V</font></span>)</span>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a'</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">b'</font> holds <br/> <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Olive" title="b3">a'</font>,<font color="Olive" title="b4">b'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c2">W</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c5">a'</font>, <font color="Maroon" title="c6">b'</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c1">V</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:58_1"/>
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a'</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b'</font> )
 ;<br/>

<a NAME="E2:58_1"/><b>then </b>
(  <a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp;  <a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c5">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp;  <a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> &amp;  <a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c6">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> &amp;  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c3">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c4">b</font></span>)</span> &amp;  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K7" title="ENS_1:func.7">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c5">a'</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c6">b'</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="ens_1.html#T27" target="_self" title="ENS_1:th.27">Th27</a></i>;<br/>
<b>hence </b><a NAME="E3:58_1"/>
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font>
 <b>by </b><i><a class="txt" href="ens_1.html#E33"><i><font color="Green" title="E23">Lm5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E2:58"/>
 <a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c2">W</font> is    <a href="cat_2.html#M3" title="CAT_2:mode.3">Subcategory</a> of  <a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c1">V</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="cat_2.html#D6" title="CAT_2:def.6">CAT_2:def 6</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:58_2"/>
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c2">W</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c1">V</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_2.html#D4" title="CAT_2:def.4">CAT_2:def 4</a><br/>

<b>thus </b><a NAME="E2:58_2"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c2">W</font></span>)</span><br/> for <font color="Olive" title="b3">a'</font>, <font color="Olive" title="b4">b'</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c1">V</font></span>)</span>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a'</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">b'</font> holds <br/> <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Olive" title="b3">a'</font>,<font color="Olive" title="b4">b'</font>
 <b>by </b><i><a class="txt" href="ens_1.html#E1:58"><i><font color="Green" title="E35">A1</font></i></a></i>;<br/>

<b>set </b><font color="Maroon" title="c3">w</font> = the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c2">W</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c4">v</font> = the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c1">V</font></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2_1"><b>now </b></a><div class="add"><a NAME="E1:58_2_1"/><i><font color="Green" title="E36">A2</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c2">W</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c2">W</font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c2">W</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p2">(<span class="default"><a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/><a NAME="E2:58_2_1"/><i><font color="Green" title="E37">A3</font></i>: 
 <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c2">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="ref" href="ens_1.html#T7" target="_self" title="ENS_1:th.7">Th7</a></i>;<br/><b>thus </b><a NAME="E3:58_2_1"/><i><font color="Green" title="E38">A4</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c2">W</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c1">V</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</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="relat_1.html#D3" title="RELAT_1:def.3">RELAT_1:def 3</a><br/>

<b>assume </b><a NAME="E1:58_2_1_1"/><i><font color="Green" title="E39">A5</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c2">W</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">m2</font>, <font color="Maroon" title="c8">m1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c2">W</font><b> such that </b><br/><a NAME="E3:58_2_1_1"/><i><font color="Green" title="E40">A6</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</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="c7">m2</font>,<font color="Maroon" title="c8">m1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="txt" href="ens_1.html#E1:58_2_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">m1'</font> = <font color="Maroon" title="c8">m1</font>, <font color="Maroon" title="c10">m2'</font> = <font color="Maroon" title="c7">m2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="ens_1.html#E2:58_2_1"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E5:58_2_1_1"/>
(  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c10">m2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c7">m2</font> &amp;  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c9">m1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c8">m1</font> &amp;  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c7">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c8">m1</font> )
 
<b>by </b><i><a class="txt" href="ens_1.html#E1:58_2_1_1"><i><font color="Green" title="E39">A5</font></i></a>, <a class="txt" href="ens_1.html#E3:58_2_1_1"><i><font color="Green" title="E40">A6</font></i></a>, <a class="ref" href="ens_1.html#D12" target="_self" title="ENS_1:def.12">Def12</a></i>;<br/>
<b>hence </b><a NAME="E6:58_2_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c1">V</font></span>)</span>
 <b>by </b><i><a class="txt" href="ens_1.html#E3:58_2_1_1"><i><font color="Green" title="E40">A6</font></i></a>, <a class="ref" href="ens_1.html#D12" target="_self" title="ENS_1:def.12">Def12</a></i>;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E4:58_2_1"/><i><font color="Green" title="E39">A7</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c2">W</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">m2</font>, <font color="Maroon" title="c7">m1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c2">W</font><b> such that </b><br/><a NAME="E6:58_2_1"/><i><font color="Green" title="E40">A8</font></i>: 
<font color="Maroon" title="c5">x</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"><font color="Maroon" title="c6">m2</font>,<font color="Maroon" title="c7">m1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="txt" href="ens_1.html#E1:58_2_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">m1'</font> = <font color="Maroon" title="c7">m1</font>, <font color="Maroon" title="c9">m2'</font> = <font color="Maroon" title="c6">m2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="ens_1.html#E2:58_2_1"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E8:58_2_1"/><i><font color="Green" title="E41">A9</font></i>: 
(  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c9">m2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c8">m1'</font> &amp;  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c7">m1</font> )
 <b>by </b><i><a class="txt" href="ens_1.html#E3:58_2_1"><i><font color="Green" title="E38">A4</font></i></a>, <a class="txt" href="ens_1.html#E4:58_2_1"><i><font color="Green" title="E39">A7</font></i></a>, <a class="txt" href="ens_1.html#E6:58_2_1"><i><font color="Green" title="E40">A8</font></i></a>, <a class="ref" href="ens_1.html#D12" target="_self" title="ENS_1:def.12">Def12</a></i>;<br/><a NAME="E9:58_2_1"/>
( <font color="Maroon" title="c7">m1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">m1'</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  &amp; <font color="Maroon" title="c6">m2</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">m2'</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  &amp;  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c7">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c8">m1'</font> &amp;  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c9">m2'</font> )
 ;<br/><a NAME="E10:58_2_1"/><b>then </b>
( <font color="Maroon" title="c6">m2</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c7">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"><span class="p3">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c6">m2</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">m2</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">m1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c9">m2'</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c8">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"><span class="p3">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c7">m1</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c6">m2</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">m2</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">m1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="ens_1.html#E8:58_2_1"><i><font color="Green" title="E41">A9</font></i></a>, <a class="ref" href="ens_1.html#D7" target="_self" title="ENS_1:def.7">Def7</a></i>;<br/><b>hence </b>the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c2">W</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> = 
<font color="Maroon" title="c9">m2'</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c8">m1'</font>
<b>by </b><i><a class="txt" href="ens_1.html#E6:58_2_1"><i><font color="Green" title="E40">A8</font></i></a>, <a class="txt" href="ens_1.html#E8:58_2_1"><i><font color="Green" title="E41">A9</font></i></a>, <a class="ref" href="ens_1.html#D12" target="_self" title="ENS_1:def.12">Def12</a></i>
<br/>.= 
the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>
<b>by </b><i><a class="txt" href="ens_1.html#E6:58_2_1"><i><font color="Green" title="E40">A8</font></i></a>, <a class="txt" href="ens_1.html#E8:58_2_1"><i><font color="Green" title="E41">A9</font></i></a>, <a class="ref" href="ens_1.html#D12" target="_self" title="ENS_1:def.12">Def12</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:58_2"/>
the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c2">W</font></span>)</span> <a href="funct_4.html#NR1" title="FUNCT_4:NR.1">&lt;=</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c1">V</font></span>)</span>
 <b>by </b><i><a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>;<br/>

<b>let </b><font color="Maroon" title="c5">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c2">W</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c6">a'</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c1">V</font></span>)</span>;<br/>



<b>assume </b><a NAME="E5:58_2"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a'</font>
 ;<br/>

<a NAME="E6:58_2"/><b>then </b>
(  <a href="ens_1.html#K5" title="ENS_1:func.5">id$</a> <span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p3">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp;  <a href="ens_1.html#K5" title="ENS_1:func.5">id$</a> <span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c6">a'</font></span>)</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"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p3">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 
;<br/>
<b>hence </b> <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c5">a</font> = 
 <a href="ens_1.html#K5" title="ENS_1:func.5">id$</a> <span class="p1">(<span class="default"><a href="ens_1.html#K14" title="ENS_1:func.14">@</a> <font color="Maroon" title="c6">a'</font></span>)</span>
<b>by </b><i><a class="ref" href="ens_1.html#D13" target="_self" title="ENS_1:def.13">Def13</a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c6">a'</font>
<b>by </b><i><a class="ref" href="ens_1.html#D13" target="_self" title="ENS_1:def.13">Def13</a></i>
;<br/><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:58"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c2">W</font></span>)</span><br/> for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c1">V</font></span>)</span>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>3</sub></font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>4</sub></font> or  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="ens_1.html#E1:58"><i><font color="Green" title="E35">A1</font></i></a></i>;<br/>


</div>
