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

<span class="kw">let </span><font color="Maroon" title="c1">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>; <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="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> holds <br/> ( <font color="Olive" title="b1">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  is  <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a>  iff <font color="Olive" title="b1">f</font> is  <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  is  <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a>  iff <font color="Maroon" title="c2">f</font> is  <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a>  )</span><br/>



<span class="kw">thus </span><a NAME="E1:18"/>
( <font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  is  <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a>  implies <font color="Maroon" title="c2">f</font> is  <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> is  <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a>  implies <font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  is  <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c3">g</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="oppcat_1.html#K2" title="OPPCAT_1:func.2">opp</a></span>)</span><span class="kw"> such that </span><a NAME="E2:18_1"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> )
 <span class="kw">and </span><br/><a NAME="E3:18_1"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span></span>)</span> &amp; <font color="Maroon" title="c3">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span></span>)</span> )
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="cat_1.html#D9" title="CAT_1:def.9">CAT_1:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font> is  <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a> </span><br/>

<span class="kw">take </span><font color="Maroon" title="c4">g9</font> =  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c3">g</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="cat_1.html#D9" title="CAT_1:def.9">CAT_1:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c4">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c4">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c2">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c4">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c4">g9</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E4:18_1"/>
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c4">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c4">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_1"><span class="lab"><font color="Green" title="E9">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c4">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c4">g9</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> )</span><br/>

<a NAME="E5:18_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c4">g9</font></span>)</span> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g9</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">g9</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g9</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#T16" target="_self" title="OPPCAT_1:th.16">Th17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:18_1"/>
( <font color="Maroon" title="c2">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c4">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c4">g9</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_1"><span class="lab"><font color="Green" title="E10">A2</font></span></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">given </span><font color="Maroon" title="c3">g</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font><span class="kw"> such that </span><a NAME="E3:18"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> )
 <span class="kw">and </span><br/><a NAME="E4:18"/><span class="lab"><font color="Green" title="E10">A4</font></span>: 
( <font color="Maroon" title="c2">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c3">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="cat_1.html#D9" title="CAT_1:def.9">CAT_1:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  is  <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a> </span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c3">g</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> 
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="cat_1.html#D9" title="CAT_1:def.9">CAT_1:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E5:18"/>
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E9">A3</font></span></a></span>; <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"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K2" title="CAT_1:func.2">id</a> <span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span></span>)</span> )</span><br/>

<a NAME="E6:18"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#T17" target="_self" title="OPPCAT_1:th.17">Th18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:18"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18"><span class="lab"><font color="Green" title="E10">A4</font></span></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>
