<?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>set </b><font color="Maroon" title="c2">M</font> =  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font>;<br/>
<b>set </b><font color="Maroon" title="c3">d</font> =  <a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <font color="Maroon" title="c1">V</font>;<br/>
<b>set </b><font color="Maroon" title="c4">c</font> =  <a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <font color="Maroon" title="c1">V</font>;<br/>
<b>set </b><font color="Maroon" title="c5">p</font> =  <a href="ens_1.html#K10" title="ENS_1:func.10">fComp</a> <font color="Maroon" title="c1">V</font>;<br/>
<b>set </b><font color="Maroon" title="c6">i</font> =  <a href="ens_1.html#K11" title="ENS_1:func.11">fId</a> <font color="Maroon" title="c1">V</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:39_1"/>
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  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K10" title="ENS_1:func.10">fComp</a> <font color="Maroon" title="c1">V</font></span>)</span> iff <span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">f</font>, <font color="Maroon" title="c8">g</font> be    <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>;<br/>

<a NAME="E1:39_1_1"/>
( <span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">g</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">g</font> &amp; <span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f</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">f</font> )
 
<b>by </b><i><a class="ref" href="ens_1.html#D10" target="_self" title="ENS_1:def.10">Def10</a>, <a class="ref" href="ens_1.html#D11" target="_self" title="ENS_1:def.11">Def11</a></i>;<br/>
<b>hence </b><a NAME="E2:39_1_1"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c7">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K10" title="ENS_1:func.10">fComp</a> <font color="Maroon" title="c1">V</font></span>)</span> iff <span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f</font> )
 <b>by </b><i><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>thus </b><a NAME="E2:39_1"/>
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  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font>  st <span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> holds <br/>( <span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K10" title="ENS_1:func.10">fComp</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> &amp; <span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K10" title="ENS_1:func.10">fComp</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">f</font>, <font color="Maroon" title="c8">g</font> be    <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>;<br/>

<b>assume </b><a NAME="E1:39_1_2"/><i><font color="Green" title="E29">A1</font></i>: 
<span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f</font>
 ;<br/>

<a NAME="E2:39_1_2"/>
( <span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">g</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">g</font> &amp; <span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f</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">f</font> )
 
<b>by </b><i><a class="ref" href="ens_1.html#D10" target="_self" title="ENS_1:def.10">Def10</a>, <a class="ref" href="ens_1.html#D11" target="_self" title="ENS_1:def.11">Def11</a></i>;<br/>
<a NAME="E3:39_1_2"/><b>then </b>
(  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c7">f</font></span>)</span> <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">f</font> &amp;  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c7">f</font></span>)</span> <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">g</font> &amp; <span class="p1">(<span class="default"><a href="ens_1.html#K10" title="ENS_1:func.10">fComp</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c7">f</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c7">f</font> &amp; <span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f</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">f</font> &amp; <span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">g</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">g</font> )
 
<b>by </b><i><a class="txt" href="ens_1.html#E1:39_1_2"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="ens_1.html#D10" target="_self" title="ENS_1:def.10">Def10</a>, <a class="ref" href="ens_1.html#D11" target="_self" title="ENS_1:def.11">Def11</a>, <a class="ref" href="ens_1.html#D12" target="_self" title="ENS_1:def.12">Def12</a>, <a class="ref" href="ens_1.html#T12" target="_self" title="ENS_1:th.12">Th12</a></i>;<br/>
<b>hence </b><a NAME="E4:39_1_2"/>
( <span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K10" title="ENS_1:func.10">fComp</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c7">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f</font> &amp; <span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K10" title="ENS_1:func.10">fComp</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c7">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">g</font> )
 <b>by </b><i><a class="ref" href="ens_1.html#D10" target="_self" title="ENS_1:def.10">Def10</a>, <a class="ref" href="ens_1.html#D11" target="_self" title="ENS_1:def.11">Def11</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:39_1"/>
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</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="c1">V</font>  st <span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> &amp; <span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> holds <br/><span class="p1">(<span class="default"><a href="ens_1.html#K10" title="ENS_1:func.10">fComp</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b3">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K10" title="ENS_1:func.10">fComp</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K10" title="ENS_1:func.10">fComp</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K10" title="ENS_1:func.10">fComp</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b3">h</font>,<font color="Olive" title="b2">g</font></span>)</span>,<font color="Olive" title="b1">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">f</font>, <font color="Maroon" title="c8">g</font>, <font color="Maroon" title="c9">h</font> be    <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>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:39_1_3"/><i><font color="Green" title="E29">A2</font></i>: 
<span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">g</font>
 <b>and </b><br/><a NAME="E2:39_1_3"/><i><font color="Green" title="E30">A3</font></i>: 
<span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f</font>
 ;<br/>

<a NAME="E3:39_1_3"/><i><font color="Green" title="E31">A4</font></i>: 
(  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c9">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">h</font> &amp;  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">g</font> &amp;  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">g</font> &amp;  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f</font> )
 
<b>by </b><i><a class="ref" href="ens_1.html#D10" target="_self" title="ENS_1:def.10">Def10</a>, <a class="ref" href="ens_1.html#D11" target="_self" title="ENS_1:def.11">Def11</a></i>;<br/>
<a NAME="E4:39_1_3"/><b>then </b><i><font color="Green" title="E32">A5</font></i>: 
(  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c7">f</font></span>)</span> <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="c9">h</font> &amp;  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c8">g</font></span>)</span> <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">g</font> )
 
<b>by </b><i><a class="txt" href="ens_1.html#E1:39_1_3"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="ens_1.html#E2:39_1_3"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="ens_1.html#T12" target="_self" title="ENS_1:th.12">Th12</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="ens_1.html#K10" title="ENS_1:func.10">fComp</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K10" title="ENS_1:func.10">fComp</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c7">f</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ens_1.html#K10" title="ENS_1:func.10">fComp</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">h</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c7">f</font></span>)</span></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#E2:39_1_3"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="ens_1.html#E3:39_1_3"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="ens_1.html#D12" target="_self" title="ENS_1:def.12">Def12</a></i>
<br/>.= 
<font color="Maroon" title="c9">h</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">g</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c7">f</font></span>)</span>
<b>by </b><i><a class="txt" href="ens_1.html#E4:39_1_3"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="ens_1.html#D12" target="_self" title="ENS_1:def.12">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c8">g</font></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c7">f</font>
<b>by </b><i><a class="txt" href="ens_1.html#E1:39_1_3"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="ens_1.html#E2:39_1_3"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="ens_1.html#E3:39_1_3"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="ens_1.html#T13" target="_self" title="ENS_1:th.13">Th13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ens_1.html#K10" title="ENS_1:func.10">fComp</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c8">g</font></span>)</span>,<font color="Maroon" title="c7">f</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#E2:39_1_3"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="ens_1.html#E3:39_1_3"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="ens_1.html#E4:39_1_3"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="ens_1.html#D12" target="_self" title="ENS_1:def.12">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ens_1.html#K10" title="ENS_1:func.10">fComp</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K10" title="ENS_1:func.10">fComp</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">h</font>,<font color="Maroon" title="c8">g</font></span>)</span>,<font color="Maroon" title="c7">f</font>
<b>by </b><i><a class="txt" href="ens_1.html#E1:39_1_3"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="ens_1.html#E3:39_1_3"><i><font color="Green" title="E31">A4</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="c7">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font>;<br/><a NAME="E4:39_1"/><i><font color="Green" title="E29">A6</font></i>: 
( <span class="p1">(<span class="default"><a href="ens_1.html#K11" title="ENS_1:func.11">fId</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K5" title="ENS_1:func.5">id$</a> <font color="Maroon" title="c7">b</font> &amp;  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K5" title="ENS_1:func.5">id$</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> &amp;  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K5" title="ENS_1:func.5">id$</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> )
 <b>by </b><i><a class="ref" href="ens_1.html#D13" target="_self" title="ENS_1:def.13">Def13</a>, <a class="ref" href="ens_1.html#T11" target="_self" title="ENS_1:th.11">Th11</a></i>;<br/><b>hence </b><a NAME="E5:39_1"/>
( <span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K11" title="ENS_1:func.11">fId</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> &amp; <span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K11" title="ENS_1:func.11">fId</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> )
 <b>by </b><i><a class="ref" href="ens_1.html#D10" target="_self" title="ENS_1:def.10">Def10</a>, <a class="ref" href="ens_1.html#D11" target="_self" title="ENS_1:def.11">Def11</a></i>;<br/><b>thus </b><a NAME="E6:39_1"/>
for <font color="Olive" title="b1">f</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="c1">V</font>  st <span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> holds <br/><span class="p1">(<span class="default"><a href="ens_1.html#K10" title="ENS_1:func.10">fComp</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K11" title="ENS_1:func.11">fId</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">b</font></span>)</span>,<font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">f</font> be    <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>;<br/>

<b>assume </b><a NAME="E1:39_1_4"/><i><font color="Green" title="E30">A7</font></i>: 
<span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font>
 ;<br/>

<a NAME="E2:39_1_4"/><i><font color="Green" title="E31">A8</font></i>: 
 <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f</font>
 
<b>by </b><i><a class="ref" href="ens_1.html#D11" target="_self" title="ENS_1:def.11">Def11</a></i>;<br/>
<a NAME="E3:39_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="ens_1.html#K5" title="ENS_1:func.5">id$</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font>
 
<b>by </b><i><a class="txt" href="ens_1.html#E1:39_1_4"><i><font color="Green" title="E30">A7</font></i></a>, <a class="ref" href="ens_1.html#T14" target="_self" title="ENS_1:th.14">Th14</a></i>;<br/>
<b>hence </b><a NAME="E4:39_1_4"/>
<span class="p1">(<span class="default"><a href="ens_1.html#K10" title="ENS_1:func.10">fComp</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K11" title="ENS_1:func.11">fId</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">b</font></span>)</span>,<font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font>
 <b>by </b><i><a class="txt" href="ens_1.html#E4:39_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="ens_1.html#E1:39_1_4"><i><font color="Green" title="E30">A7</font></i></a>, <a class="txt" href="ens_1.html#E2:39_1_4"><i><font color="Green" title="E31">A8</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="c8">g</font> be    <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>;<br/><b>assume </b><a NAME="E7:39_1"/><i><font color="Green" title="E30">A9</font></i>: 
<span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font>
 ;<br/><a NAME="E8:39_1"/><i><font color="Green" title="E31">A10</font></i>: 
 <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">g</font>
 <b>by </b><i><a class="ref" href="ens_1.html#D10" target="_self" title="ENS_1:def.10">Def10</a></i>;<br/><a NAME="E9:39_1"/><b>then </b>
<font color="Maroon" title="c8">g</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K5" title="ENS_1:func.5">id$</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font>
 <b>by </b><i><a class="txt" href="ens_1.html#E7:39_1"><i><font color="Green" title="E30">A9</font></i></a>, <a class="ref" href="ens_1.html#T14" target="_self" title="ENS_1:th.14">Th14</a></i>;<br/><b>hence </b><a NAME="E10:39_1"/>
<span class="p1">(<span class="default"><a href="ens_1.html#K10" title="ENS_1:func.10">fComp</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">g</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K11" title="ENS_1:func.11">fId</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font>
 <b>by </b><i><a class="txt" href="ens_1.html#E4:39_1"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="ens_1.html#E7:39_1"><i><font color="Green" title="E30">A9</font></i></a>, <a class="txt" href="ens_1.html#E8:39_1"><i><font color="Green" title="E31">A10</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>hence </b><a NAME="E2:39"/>
 <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c1">V</font>,<span class="p1">(<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="p1">(<span class="default"><a href="ens_1.html#K8" title="ENS_1:func.8">fDom</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K9" title="ENS_1:func.9">fCod</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K10" title="ENS_1:func.10">fComp</a> <font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K11" title="ENS_1:func.11">fId</a> <font color="Maroon" title="c1">V</font></span>)</span> #) is   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>
 <b>by </b><i><a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a></i>;<br/>


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