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<b>let </b><font color="Maroon" title="c1">I</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/><b>let </b><font color="Maroon" title="c3">F</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">I</font>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C</font>;<br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>now </b></a><div class="add"><a NAME="E1:16_1"/>
( <font color="Maroon" title="c2">C</font> <a href="oppcat_1.html#K2" title="OPPCAT_1:func.2">opp</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c2">C</font>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C</font>,the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c2">C</font>,the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c2">C</font>,<span class="p1">(<span class="default"><a href="oppcat_1.html#K1" title="OPPCAT_1:func.1">~</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c2">C</font></span>)</span>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c2">C</font> #) &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="oppcat_1.html#K2" title="OPPCAT_1:func.2">opp</a> </span>)</span> <a href="oppcat_1.html#K2" title="OPPCAT_1:func.2">opp</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="oppcat_1.html#K2" title="OPPCAT_1:func.2">opp</a> </span>)</span>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="oppcat_1.html#K2" title="OPPCAT_1:func.2">opp</a> </span>)</span>,the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="oppcat_1.html#K2" title="OPPCAT_1:func.2">opp</a> </span>)</span>,the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="oppcat_1.html#K2" title="OPPCAT_1:func.2">opp</a> </span>)</span>,<span class="p1">(<span class="default"><a href="oppcat_1.html#K1" title="OPPCAT_1:func.1">~</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="oppcat_1.html#K2" title="OPPCAT_1:func.2">opp</a> </span>)</span></span>)</span>,the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="oppcat_1.html#K2" title="OPPCAT_1:func.2">opp</a> </span>)</span> #) )
 <b>by </b><i><a class="ref" href="oppcat_1.html#D1" title="OPPCAT_1:def.1">OPPCAT_1:def 1</a></i>;<br/><b>hence </b><a NAME="E2:16_1"/>
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="oppcat_1.html#K2" title="OPPCAT_1:func.2">opp</a> </span>)</span> <a href="oppcat_1.html#K2" title="OPPCAT_1:func.2">opp</a> </span>)</span>
 ;<br/><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E3:16_1"/><i><font color="Green" title="E12">A1</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="cat_3.html#K4" title="CAT_3:func.4">opp</a> </span>)</span> <a href="cat_3.html#K4" title="CAT_3:func.4">opp</a> </span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="cat_3.html#K4" title="CAT_3:func.4">opp</a> </span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> 
<b>by </b><i><a class="ref" href="cat_3.html#D5" target="_self" title="CAT_3:def.5">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> 
<b>by </b><i><a class="txt" href="cat_3.html#E3:16_1"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="cat_3.html#D5" target="_self" title="CAT_3:def.5">Def5</a></i>
<br/>.= 
<font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="ref" href="oppcat_1.html#T6" title="OPPCAT_1:th.6">OPPCAT_1:6</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:16"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="cat_3.html#K4" title="CAT_3:func.4">opp</a> </span>)</span> <a href="cat_3.html#K4" title="CAT_3:func.4">opp</a>  <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="ref" href="cat_3.html#T1" target="_self" title="CAT_3:th.1">Th1</a></i>;<br/>


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