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

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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></span><br/>

<span class="kw">set </span><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/>
<span class="kw">set </span><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/>
<span class="kw">set </span><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/>
<span class="kw">set </span><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/>
<span class="kw">set </span><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="38_1"><span class="kw">now </span></a><div class="add"><span class="kw">thus </span><a NAME="E1:38_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> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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  <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> ) ) &amp; (  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>) ) &amp; (  for <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="c1">V</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_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="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; (  for <font color="Olive" title="b2">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="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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="Olive" title="b1">b</font></span>)</span>,<font color="Olive" title="b2">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> ) &amp; (  for <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> <font color="Olive" title="b1">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> (<font color="Olive" title="b2">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="Olive" title="b1">b</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) ) ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> )</span><br/>

<a NAME="E1:38_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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D9" target="_self" title="ENS_1:def.9">Def10</a>, <a class="ref" href="ens_1.html#D10" target="_self" title="ENS_1:def.10">Def11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:38_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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D11" target="_self" title="ENS_1:def.11">Def12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E2:38_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> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  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>) ) &amp; (  for <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="c1">V</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_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="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; (  for <font color="Olive" title="b2">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="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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="Olive" title="b1">b</font></span>)</span>,<font color="Olive" title="b2">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> ) &amp; (  for <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> <font color="Olive" title="b1">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> (<font color="Olive" title="b2">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="Olive" title="b1">b</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) ) ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> implies ( <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> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:38_1_2"/><span class="lab"><font color="Green" title="E29">A1</font></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> <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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> )</span><br/>

<a NAME="E2:38_1_2"/><span class="lab"><font color="Green" title="E30">A2</font></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> <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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D9" target="_self" title="ENS_1:def.9">Def10</a>, <a class="ref" href="ens_1.html#D10" target="_self" title="ENS_1:def.10">Def11</a></span>;<br/>
<a NAME="E3:38_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A3</font></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="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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1_2"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="ens_1.html#D11" target="_self" title="ENS_1:def.11">Def12</a></span>;<br/>
<a NAME="E4:38_1_2"/><span class="lab"><font color="Green" title="E32">A4</font></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> <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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D9" target="_self" title="ENS_1:def.9">Def10</a>, <a class="ref" href="ens_1.html#D10" target="_self" title="ENS_1:def.10">Def11</a></span>;<br/>
<a NAME="E5:38_1_2"/>
(  <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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1_2"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E2:38_1_2"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="ens_1.html#T12" target="_self" title="ENS_1:th.12">Th12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:38_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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1_2"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E4:38_1_2"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="ens_1.html#D9" target="_self" title="ENS_1:def.9">Def10</a>, <a class="ref" href="ens_1.html#D10" target="_self" title="ENS_1:def.10">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E3:38_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>)
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <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="c1">V</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_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="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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="Olive" title="b1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; (  for <font color="Olive" title="b2">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="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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="Olive" title="b1">b</font></span>)</span>,<font color="Olive" title="b2">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> ) &amp; (  for <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> <font color="Olive" title="b1">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> (<font color="Olive" title="b2">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="Olive" title="b1">b</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &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="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> implies <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>) <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="Maroon" title="c9">h</font>,<font color="Maroon" title="c8">g</font>)</span>)</span>,<font color="Maroon" title="c7">f</font>) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:38_1_3"/><span class="lab"><font color="Green" title="E29">A5</font></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> <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>
 <span class="kw">and </span><br/><a NAME="E2:38_1_3"/><span class="lab"><font color="Green" title="E30">A6</font></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> <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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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>) <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="Maroon" title="c9">h</font>,<font color="Maroon" title="c8">g</font>)</span>)</span>,<font color="Maroon" title="c7">f</font>)</span><br/>

<a NAME="E3:38_1_3"/><span class="lab"><font color="Green" title="E31">A7</font></span>: 
(  <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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D9" target="_self" title="ENS_1:def.9">Def10</a>, <a class="ref" href="ens_1.html#D10" target="_self" title="ENS_1:def.10">Def11</a></span>;<br/>
<a NAME="E4:38_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A8</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1_3"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="ref" href="ens_1.html#T12" target="_self" title="ENS_1:th.12">Th12</a></span>;<br/>
<a NAME="E5:38_1_3"/><span class="lab"><font color="Green" title="E33">A9</font></span>: 
(  <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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D9" target="_self" title="ENS_1:def.9">Def10</a>, <a class="ref" href="ens_1.html#D10" target="_self" title="ENS_1:def.10">Def11</a></span>;<br/>
<a NAME="E6:38_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A10</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1_3"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E2:38_1_3"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E3:38_1_3"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="ref" href="ens_1.html#T12" target="_self" title="ENS_1:th.12">Th12</a></span>;<br/>
<span class="kw">thus </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="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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38_1_3"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E5:38_1_3"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="ref" href="ens_1.html#D11" target="_self" title="ENS_1:def.11">Def12</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:38_1_3"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="ref" href="ens_1.html#D11" target="_self" title="ENS_1:def.11">Def12</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1_3"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E2:38_1_3"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E3:38_1_3"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E5:38_1_3"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="ref" href="ens_1.html#T13" target="_self" title="ENS_1:th.13">Th13</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38_1_3"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E5:38_1_3"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="txt" href="#E4:38_1_3"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="ref" href="ens_1.html#D11" target="_self" title="ENS_1:def.11">Def12</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1_3"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E3:38_1_3"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="ref" href="ens_1.html#D11" target="_self" title="ENS_1:def.11">Def12</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; (  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> ) &amp; (  for <font color="Olive" title="b1">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="b1">g</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> (<font color="Olive" title="b1">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="Olive" title="b1">g</font> ) )</span><br/><a NAME="E4:38_1"/><span class="lab"><font color="Green" title="E29">A11</font></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> <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$</a> <font color="Maroon" title="c7">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D12" target="_self" title="ENS_1:def.12">Def13</a></span>;<br/><a NAME="E5:38_1"/><span class="lab"><font color="Green" title="E30">A12</font></span>: 
 <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K5" title="ENS_1:func.5">id$</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T11" target="_self" title="ENS_1:th.11">Th11</a></span>;<br/><a NAME="E6:38_1"/><span class="lab"><font color="Green" title="E31">A13</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T11" target="_self" title="ENS_1:th.11">Th11</a></span>;<br/><span class="kw">hence </span><a NAME="E7:38_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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38_1"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="txt" href="#E5:38_1"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="ref" href="ens_1.html#D9" target="_self" title="ENS_1:def.9">Def10</a>, <a class="ref" href="ens_1.html#D10" target="_self" title="ENS_1:def.10">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  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> ) &amp; (  for <font color="Olive" title="b1">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="b1">g</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> (<font color="Olive" title="b1">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="Olive" title="b1">g</font> ) )</span><br/><span class="kw">thus </span><a NAME="E8:38_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>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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  <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="b1">g</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> (<font color="Olive" title="b1">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="Olive" title="b1">g</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> implies <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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:38_1_4"/><span class="lab"><font color="Green" title="E32">A14</font></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> <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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<a NAME="E2:38_1_4"/><span class="lab"><font color="Green" title="E33">A15</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D10" target="_self" title="ENS_1:def.10">Def11</a></span>;<br/>
<a NAME="E3:38_1_4"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1_4"><span class="lab"><font color="Green" title="E32">A14</font></span></a>, <a class="ref" href="ens_1.html#T14" target="_self" title="ENS_1:th.14">Th14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:38_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38_1"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="txt" href="#E6:38_1"><span class="lab"><font color="Green" title="E31">A13</font></span></a>, <a class="txt" href="#E1:38_1_4"><span class="lab"><font color="Green" title="E32">A14</font></span></a>, <a class="txt" href="#E2:38_1_4"><span class="lab"><font color="Green" title="E33">A15</font></span></a>, <a class="ref" href="ens_1.html#D11" target="_self" title="ENS_1:def.11">Def12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> implies <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> )</span><br/><span class="kw">assume </span><a NAME="E9:38_1"/><span class="lab"><font color="Green" title="E32">A16</font></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> <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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/><a NAME="E10:38_1"/><span class="lab"><font color="Green" title="E33">A17</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#D9" target="_self" title="ENS_1:def.9">Def10</a></span>;<br/><a NAME="E11:38_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:38_1"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="ref" href="ens_1.html#T14" target="_self" title="ENS_1:th.14">Th14</a></span>;<br/><span class="kw">hence </span><a NAME="E12:38_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38_1"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="txt" href="#E5:38_1"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="txt" href="#E9:38_1"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="txt" href="#E10:38_1"><span class="lab"><font color="Green" title="E33">A17</font></span></a>, <a class="ref" href="ens_1.html#D11" target="_self" title="ENS_1:def.11">Def12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:38"/>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#D5" title="CAT_1:def.5">CAT_1:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
