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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">c</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font><br/> for <font color="Olive" title="b2">p1</font>, <font color="Olive" title="b3">p2</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">c</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Olive" title="b2">p1</font>,<font color="Olive" title="b3">p2</font> iff <font color="Olive" title="b1">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a>  <a href="cat_3.html#R4" title="CAT_3:pred.4">is_a_coproduct_wrt</a> <font color="Olive" title="b2">p1</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> ,<font color="Olive" title="b3">p2</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  )</span><br/><b>let </b><font color="Maroon" title="c2">c</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</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="Maroon" title="c2">c</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font> iff <font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a>  <a href="cat_3.html#R4" title="CAT_3:pred.4">is_a_coproduct_wrt</a> <font color="Olive" title="b1">p1</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> ,<font color="Olive" title="b2">p2</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  )</span><br/><b>let </b><font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c4">p2</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">c</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> iff <font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a>  <a href="cat_3.html#R4" title="CAT_3:pred.4">is_a_coproduct_wrt</a> <font color="Maroon" title="c3">p1</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> ,<font color="Maroon" title="c4">p2</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  )</span><br/>







<b>set </b><font color="Maroon" title="c5">i1</font> = <font color="Maroon" title="c3">p1</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> ;<br/>
<b>set </b><font color="Maroon" title="c6">i2</font> = <font color="Maroon" title="c4">p2</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> ;<br/>
<b>thus </b><a NAME="E1:96"/>
( <font color="Maroon" title="c2">c</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> implies <font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a>  <a href="cat_3.html#R4" title="CAT_3:pred.4">is_a_coproduct_wrt</a> <font color="Maroon" title="c3">p1</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> ,<font color="Maroon" title="c4">p2</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a>  <a href="cat_3.html#R4" title="CAT_3:pred.4">is_a_coproduct_wrt</a> <font color="Maroon" title="c3">p1</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> ,<font color="Maroon" title="c4">p2</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  implies <font color="Maroon" title="c2">c</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:96_1"/><i><font color="Green" title="E56">A1</font></i>: 
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c3">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c4">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> )
 <b>and </b><br/><a NAME="E2:96_1"/><i><font color="Green" title="E57">A2</font></i>: 
for <font color="Olive" title="b1">d</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font><br/> for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive" title="b1">d</font>,<span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c3">p1</font></span>)</span> &amp; <font color="Olive" title="b3">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive" title="b1">d</font>,<span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c4">p2</font></span>)</span> holds <br/> ex <font color="Olive" title="b4">h</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> st <br/>( <font color="Olive" title="b4">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive" title="b1">d</font>,<font color="Maroon" title="c2">c</font> &amp; ( for <font color="Olive" title="b5">k</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive" title="b1">d</font>,<font color="Maroon" title="c2">c</font> holds <br/>( ( <font color="Maroon" title="c3">p1</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> &amp; <font color="Maroon" title="c4">p2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> ) iff <font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">k</font> ) ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_3.html#D16" target="_self" title="CAT_3:def.16">CAT_3:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a>  <a href="cat_3.html#R4" title="CAT_3:pred.4">is_a_coproduct_wrt</a> <font color="Maroon" title="c3">p1</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> ,<font color="Maroon" title="c4">p2</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span><br/>

<a NAME="E3:96_1"/>
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c3">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a>  &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c4">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a>  )
 
<b>by </b><i><a class="txt" href="#E1:96_1"><i><font color="Green" title="E56">A1</font></i></a>, <a class="ref" href="oppcat_1.html#D2" title="OPPCAT_1:def.2">OPPCAT_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:96_1"/><i><font color="Green" title="E58">A3</font></i>: 
(  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p1</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> <font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a>  &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p2</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> <font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a>  )
 <b>by </b><i><a class="ref" href="oppcat_1.html#T9" title="OPPCAT_1:th.9">OPPCAT_1:9</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_3.html#D19" target="_self" title="CAT_3:def.19">CAT_3:def 19</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">d</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</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><br/> for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">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>  st <font color="Olive" title="b2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</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="c3">p1</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span></span>)</span>,<font color="Olive" title="b1">d</font> &amp; <font color="Olive" title="b3">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</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="c4">p2</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span></span>)</span>,<font color="Olive" title="b1">d</font> holds <br/> ex <font color="Olive" title="b4">h</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> st <br/>( <font color="Olive" title="b4">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Olive" title="b1">d</font> &amp; ( for <font color="Olive" title="b5">k</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>  st <font color="Olive" title="b5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Olive" title="b1">d</font> holds <br/>( ( <font color="Olive" title="b5">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p1</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> <font color="Olive" title="b2">f</font> &amp; <font color="Olive" title="b5">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p2</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> <font color="Olive" title="b3">g</font> ) iff <font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">k</font> ) ) )</span><br/>

<b>let </b><font color="Maroon" title="c7">d</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">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>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</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="c3">p1</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span></span>)</span>,<font color="Maroon" title="c7">d</font> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</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="c4">p2</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span></span>)</span>,<font color="Maroon" title="c7">d</font> holds <br/> ex <font color="Olive" title="b3">h</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> st <br/>( <font color="Olive" title="b3">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Maroon" title="c7">d</font> &amp; ( for <font color="Olive" title="b4">k</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>  st <font color="Olive" title="b4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Maroon" title="c7">d</font> holds <br/>( ( <font color="Olive" title="b4">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p1</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> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b4">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p2</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> <font color="Olive" title="b2">g</font> ) iff <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">k</font> ) ) )</span><br/><b>let </b><font color="Maroon" title="c8">f</font>, <font color="Maroon" title="c9">g</font> be   <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</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="c3">p1</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span></span>)</span>,<font color="Maroon" title="c7">d</font> &amp; <font color="Maroon" title="c9">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</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="c4">p2</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span></span>)</span>,<font color="Maroon" title="c7">d</font> implies  ex <font color="Olive" title="b1">h</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> st <br/>( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Maroon" title="c7">d</font> &amp; ( for <font color="Olive" title="b2">k</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>  st <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Maroon" title="c7">d</font> holds <br/>( ( <font color="Olive" title="b2">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p1</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> <font color="Maroon" title="c8">f</font> &amp; <font color="Olive" title="b2">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p2</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> <font color="Maroon" title="c9">g</font> ) iff <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">k</font> ) ) ) )</span><br/>



<b>assume </b><a NAME="E5:96_1"/>
( <font color="Maroon" title="c8">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</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="c3">p1</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span></span>)</span>,<font color="Maroon" title="c7">d</font> &amp; <font color="Maroon" title="c9">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</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="c4">p2</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span></span>)</span>,<font color="Maroon" title="c7">d</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">h</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> st <br/>( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Maroon" title="c7">d</font> &amp; ( for <font color="Olive" title="b2">k</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>  st <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Maroon" title="c7">d</font> holds <br/>( ( <font color="Olive" title="b2">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p1</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> <font color="Maroon" title="c8">f</font> &amp; <font color="Olive" title="b2">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p2</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> <font color="Maroon" title="c9">g</font> ) iff <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">k</font> ) ) )</span><br/>

<a NAME="E6:96_1"/><b>then </b>
(  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <font color="Maroon" title="c7">d</font></span>)</span>,<span class="p1">(<span class="default"><a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <span class="p2">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p1</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span></span>)</span></span>)</span> &amp;  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <font color="Maroon" title="c7">d</font></span>)</span>,<span class="p1">(<span class="default"><a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <span class="p2">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p2</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T14" title="OPPCAT_1:th.14">OPPCAT_1:14</a></i>;<br/>
<a NAME="E7:96_1"/><b>then </b>
(  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <font color="Maroon" title="c7">d</font></span>)</span>,<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"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p1</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span></span>)</span></span>)</span> &amp;  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <font color="Maroon" title="c7">d</font></span>)</span>,<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"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">p2</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T12" title="OPPCAT_1:th.12">OPPCAT_1:12</a></i>;<br/>
<a NAME="E8:96_1"/><b>then </b>
(  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <font color="Maroon" title="c7">d</font></span>)</span>,<span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c3">p1</font></span>)</span> &amp;  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <font color="Maroon" title="c7">d</font></span>)</span>,<span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c4">p2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T7" title="OPPCAT_1:th.7">OPPCAT_1:7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">h</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E10:96_1"/><i><font color="Green" title="E59">A4</font></i>: 
<font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <font color="Maroon" title="c7">d</font></span>)</span>,<font color="Maroon" title="c2">c</font>
 <b>and </b><br/><a NAME="E11:96_1"/><i><font color="Green" title="E60">A5</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <font color="Maroon" title="c7">d</font></span>)</span>,<font color="Maroon" title="c2">c</font> holds <br/>( ( <font color="Maroon" title="c3">p1</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c8">f</font> &amp; <font color="Maroon" title="c4">p2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c9">g</font> ) iff <font color="Maroon" title="c10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> )
 <b>by </b><i><a class="txt" href="#E2:96_1"><i><font color="Green" title="E57">A2</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c10">h</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> 
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">h</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Maroon" title="c7">d</font> &amp; ( for <font color="Olive" title="b1">k</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>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Maroon" title="c7">d</font> holds <br/>( ( <font color="Olive" title="b1">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p1</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> <font color="Maroon" title="c8">f</font> &amp; <font color="Olive" title="b1">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p2</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> <font color="Maroon" title="c9">g</font> ) iff <font color="Maroon" title="c10">h</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> ) ) )</span><br/>

<a NAME="E12:96_1"/>
<font color="Maroon" title="c10">h</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <font color="Maroon" title="c7">d</font></span>)</span> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="#E10:96_1"><i><font color="Green" title="E59">A4</font></i></a>, <a class="ref" href="oppcat_1.html#T13" title="OPPCAT_1:th.13">OPPCAT_1:13</a></i>;<br/>
<b>hence </b><a NAME="E13:96_1"/>
<font color="Maroon" title="c10">h</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Maroon" title="c7">d</font>
 <b>by </b><i><a class="ref" href="oppcat_1.html#T5" title="OPPCAT_1:th.5">OPPCAT_1:5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">k</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>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Maroon" title="c7">d</font> holds <br/>( ( <font color="Olive" title="b1">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p1</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> <font color="Maroon" title="c8">f</font> &amp; <font color="Olive" title="b1">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p2</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> <font color="Maroon" title="c9">g</font> ) iff <font color="Maroon" title="c10">h</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> )</span><br/>

<b>let </b><font color="Maroon" title="c11">k</font> be   <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Maroon" title="c7">d</font> implies ( ( <font color="Maroon" title="c11">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p1</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> <font color="Maroon" title="c8">f</font> &amp; <font color="Maroon" title="c11">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p2</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> <font color="Maroon" title="c9">g</font> ) iff <font color="Maroon" title="c10">h</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font> ) )</span><br/>

<b>assume </b><a NAME="E14:96_1"/><i><font color="Green" title="E61">A6</font></i>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Maroon" title="c7">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c11">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p1</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> <font color="Maroon" title="c8">f</font> &amp; <font color="Maroon" title="c11">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p2</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> <font color="Maroon" title="c9">g</font> ) iff <font color="Maroon" title="c10">h</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font> )</span><br/>

<a NAME="E15:96_1"/><b>then </b>
 <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <font color="Maroon" title="c7">d</font></span>)</span>,<span class="p1">(<span class="default"><a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T14" title="OPPCAT_1:th.14">OPPCAT_1:14</a></i>;<br/>
<a NAME="E16:96_1"/><b>then </b>
 <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <font color="Maroon" title="c7">d</font></span>)</span>,<font color="Maroon" title="c2">c</font>
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T4" title="OPPCAT_1:th.4">OPPCAT_1:4</a></i>;<br/>
<a NAME="E17:96_1"/><b>then </b>
( ( <font color="Maroon" title="c3">p1</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c8">f</font> &amp; <font color="Maroon" title="c4">p2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c9">g</font> ) iff <font color="Maroon" title="c10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c11">k</font> )
 
<b>by </b><i><a class="txt" href="#E11:96_1"><i><font color="Green" title="E60">A5</font></i></a></i>;<br/>
<a NAME="E18:96_1"/><b>then </b>
( ( <font color="Maroon" title="c3">p1</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  &amp; <font color="Maroon" title="c4">p2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  ) iff <font color="Maroon" title="c10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  )
 
<b>by </b><i><a class="ref" href="oppcat_1.html#D5" title="OPPCAT_1:def.5">OPPCAT_1:def 5</a></i>;<br/>
<a NAME="E19:96_1"/><b>then </b>
( ( <font color="Maroon" title="c3">p1</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> &amp; <font color="Maroon" title="c4">p2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> ) iff <font color="Maroon" title="c10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font> )
 
<b>by </b><i><a class="ref" href="oppcat_1.html#D4" title="OPPCAT_1:def.4">OPPCAT_1:def 4</a></i>;<br/>
<a NAME="E20:96_1"/><b>then </b>
( ( <span class="p1">(<span class="default"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p1</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> &amp; <span class="p1">(<span class="default"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p2</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> implies <font color="Maroon" title="c10">h</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font> ) &amp; ( <font color="Maroon" title="c10">h</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font> implies ( <span class="p1">(<span class="default"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p1</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> &amp; <span class="p1">(<span class="default"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p2</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> ) ) &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a>  )
 
<b>by </b><i><a class="txt" href="#E14:96_1"><i><font color="Green" title="E61">A6</font></i></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a>, <a class="ref" href="oppcat_1.html#T7" title="OPPCAT_1:th.7">OPPCAT_1:7</a>, <a class="ref" href="oppcat_1.html#D4" title="OPPCAT_1:def.4">OPPCAT_1:def 4</a></i>;<br/>
<a NAME="E21:96_1"/><b>then </b>
( (  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p1</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> &amp;  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p2</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> ) iff <font color="Maroon" title="c10">h</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font> )
 
<b>by </b><i><a class="txt" href="#E4:96_1"><i><font color="Green" title="E58">A3</font></i></a>, <a class="ref" href="oppcat_1.html#T19" title="OPPCAT_1:th.19">OPPCAT_1:19</a></i>;<br/>
<a NAME="E22:96_1"/><b>then </b>
( ( <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p1</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span></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> <font color="Maroon" title="c8">f</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p2</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span></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> <font color="Maroon" title="c9">g</font> ) iff <font color="Maroon" title="c10">h</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font> )
 
<b>by </b><i><a class="ref" href="oppcat_1.html#D5" title="OPPCAT_1:def.5">OPPCAT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E23:96_1"/>
( ( <font color="Maroon" title="c11">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p1</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> <font color="Maroon" title="c8">f</font> &amp; <font color="Maroon" title="c11">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p2</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> <font color="Maroon" title="c9">g</font> ) iff <font color="Maroon" title="c10">h</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font> )
 <b>by </b><i><a class="ref" href="oppcat_1.html#D4" title="OPPCAT_1:def.4">OPPCAT_1:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><b>that </b><br/><a NAME="E2:96"/><i><font color="Green" title="E56">A7</font></i>: 
(  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p1</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> <font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a>  &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p2</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> <font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a>  )
 <b>and </b><br/><a NAME="E3:96"/><i><font color="Green" title="E57">A8</font></i>: 
for <font color="Olive" title="b1">d</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</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><br/> for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">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>  st <font color="Olive" title="b2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</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="c3">p1</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span></span>)</span>,<font color="Olive" title="b1">d</font> &amp; <font color="Olive" title="b3">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</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="c4">p2</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span></span>)</span>,<font color="Olive" title="b1">d</font> holds <br/> ex <font color="Olive" title="b4">h</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> st <br/>( <font color="Olive" title="b4">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Olive" title="b1">d</font> &amp; ( for <font color="Olive" title="b5">k</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>  st <font color="Olive" title="b5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Olive" title="b1">d</font> holds <br/>( ( <font color="Olive" title="b5">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p1</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> <font color="Olive" title="b2">f</font> &amp; <font color="Olive" title="b5">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p2</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> <font color="Olive" title="b3">g</font> ) iff <font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">k</font> ) ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_3.html#D19" target="_self" title="CAT_3:def.19">CAT_3:def 19</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">c</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font></span><br/>

<a NAME="E4:96"/>
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c3">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a>  &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c4">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a>  )
 
<b>by </b><i><a class="txt" href="#E2:96"><i><font color="Green" title="E56">A7</font></i></a>, <a class="ref" href="oppcat_1.html#T9" title="OPPCAT_1:th.9">OPPCAT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E5:96"/><i><font color="Green" title="E58">A9</font></i>: 
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c3">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c4">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> )
 <b>by </b><i><a class="ref" href="oppcat_1.html#D2" title="OPPCAT_1:def.2">OPPCAT_1:def 2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_3.html#D16" target="_self" title="CAT_3:def.16">CAT_3:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">d</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font><br/> for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive" title="b1">d</font>,<span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c3">p1</font></span>)</span> &amp; <font color="Olive" title="b3">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive" title="b1">d</font>,<span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c4">p2</font></span>)</span> holds <br/> ex <font color="Olive" title="b4">h</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> st <br/>( <font color="Olive" title="b4">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive" title="b1">d</font>,<font color="Maroon" title="c2">c</font> &amp; ( for <font color="Olive" title="b5">k</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive" title="b1">d</font>,<font color="Maroon" title="c2">c</font> holds <br/>( ( <font color="Maroon" title="c3">p1</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> &amp; <font color="Maroon" title="c4">p2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> ) iff <font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">k</font> ) ) )</span><br/>

<b>let </b><font color="Maroon" title="c7">d</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c7">d</font>,<span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c3">p1</font></span>)</span> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c7">d</font>,<span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c4">p2</font></span>)</span> holds <br/> ex <font color="Olive" title="b3">h</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> st <br/>( <font color="Olive" title="b3">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c2">c</font> &amp; ( for <font color="Olive" title="b4">k</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c2">c</font> holds <br/>( ( <font color="Maroon" title="c3">p1</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp; <font color="Maroon" title="c4">p2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) iff <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">k</font> ) ) )</span><br/><b>let </b><font color="Maroon" title="c8">f</font>, <font color="Maroon" title="c9">g</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c7">d</font>,<span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c3">p1</font></span>)</span> &amp; <font color="Maroon" title="c9">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c7">d</font>,<span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c4">p2</font></span>)</span> implies  ex <font color="Olive" title="b1">h</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> st <br/>( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c2">c</font> &amp; ( for <font color="Olive" title="b2">k</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c2">c</font> holds <br/>( ( <font color="Maroon" title="c3">p1</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> &amp; <font color="Maroon" title="c4">p2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> ) iff <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">k</font> ) ) ) )</span><br/>





<b>assume </b><a NAME="E6:96"/>
( <font color="Maroon" title="c8">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c7">d</font>,<span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c3">p1</font></span>)</span> &amp; <font color="Maroon" title="c9">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c7">d</font>,<span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c4">p2</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">h</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> st <br/>( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c2">c</font> &amp; ( for <font color="Olive" title="b2">k</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c2">c</font> holds <br/>( ( <font color="Maroon" title="c3">p1</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> &amp; <font color="Maroon" title="c4">p2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> ) iff <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">k</font> ) ) )</span><br/>

<a NAME="E7:96"/><b>then </b>
( <font color="Maroon" title="c8">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">d</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span> &amp; <font color="Maroon" title="c9">g</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c4">p2</font></span>)</span> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">d</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T13" title="OPPCAT_1:th.13">OPPCAT_1:13</a></i>;<br/>
<a NAME="E8:96"/><b>then </b>
( <font color="Maroon" title="c8">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</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="c3">p1</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">d</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span> &amp; <font color="Maroon" title="c9">g</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</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="c4">p2</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">d</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T11" title="OPPCAT_1:th.11">OPPCAT_1:11</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">h</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><b> such that </b><br/><a NAME="E10:96"/><i><font color="Green" title="E59">A10</font></i>: 
<font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">d</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>
 <b>and </b><br/><a NAME="E11:96"/><i><font color="Green" title="E60">A11</font></i>: 
for <font color="Olive" title="b1">k</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>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">d</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span> holds <br/>( ( <font color="Olive" title="b1">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p1</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> <font color="Maroon" title="c8">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  &amp; <font color="Olive" title="b1">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p2</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> <font color="Maroon" title="c9">g</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  ) iff <font color="Maroon" title="c10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> )
 <b>by </b><i><a class="txt" href="#E3:96"><i><font color="Green" title="E57">A8</font></i></a></i>;<br/>
<b>take </b>
 <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c10">h</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c2">c</font> &amp; ( for <font color="Olive" title="b1">k</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c2">c</font> holds <br/>( ( <font color="Maroon" title="c3">p1</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> &amp; <font color="Maroon" title="c4">p2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> ) iff  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> ) ) )</span><br/>

<a NAME="E12:96"/>
<font color="Maroon" title="c10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c10">h</font></span>)</span> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> 
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T8" title="OPPCAT_1:th.8">OPPCAT_1:8</a></i>;<br/>
<b>hence </b><a NAME="E13:96"/>
 <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c2">c</font>
 <b>by </b><i><a class="txt" href="#E10:96"><i><font color="Green" title="E59">A10</font></i></a>, <a class="ref" href="oppcat_1.html#T13" title="OPPCAT_1:th.13">OPPCAT_1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">k</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c2">c</font> holds <br/>( ( <font color="Maroon" title="c3">p1</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> &amp; <font color="Maroon" title="c4">p2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> ) iff  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> )</span><br/>

<b>let </b><font color="Maroon" title="c11">k</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c2">c</font> implies ( ( <font color="Maroon" title="c3">p1</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> &amp; <font color="Maroon" title="c4">p2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> ) iff  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font> ) )</span><br/>

<b>assume </b><a NAME="E14:96"/><i><font color="Green" title="E61">A12</font></i>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c2">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c3">p1</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> &amp; <font color="Maroon" title="c4">p2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> ) iff  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font> )</span><br/>

<a NAME="E15:96"/><b>then </b>
<font color="Maroon" title="c11">k</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">d</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T13" title="OPPCAT_1:th.13">OPPCAT_1:13</a></i>;<br/>
<a NAME="E16:96"/><b>then </b>
( ( <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</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">p1</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> <font color="Maroon" title="c8">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</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">p2</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> <font color="Maroon" title="c9">g</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  ) iff <font color="Maroon" title="c10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  )
 
<b>by </b><i><a class="txt" href="#E11:96"><i><font color="Green" title="E60">A11</font></i></a></i>;<br/>
<a NAME="E17:96"/><b>then </b>
( ( <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</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">p1</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> <font color="Maroon" title="c8">f</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</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">p2</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> <font color="Maroon" title="c9">g</font> implies <font color="Maroon" title="c10">h</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  ) &amp; ( <font color="Maroon" title="c10">h</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  implies ( <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</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">p1</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> <font color="Maroon" title="c8">f</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</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">p2</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> <font color="Maroon" title="c9">g</font> ) ) &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> )
 
<b>by </b><i><a class="txt" href="#E14:96"><i><font color="Green" title="E61">A12</font></i></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a>, <a class="ref" href="oppcat_1.html#D4" title="OPPCAT_1:def.4">OPPCAT_1:def 4</a></i>;<br/>
<a NAME="E18:96"/><b>then </b>
( ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">p1</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c11">k</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> <font color="Maroon" title="c8">f</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">p2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c11">k</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> <font color="Maroon" title="c9">g</font> ) iff <font color="Maroon" title="c10">h</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font> )
 
<b>by </b><i><a class="txt" href="#E5:96"><i><font color="Green" title="E58">A9</font></i></a>, <a class="ref" href="oppcat_1.html#T17" title="OPPCAT_1:th.17">OPPCAT_1:17</a>, <a class="ref" href="oppcat_1.html#D4" title="OPPCAT_1:def.4">OPPCAT_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E19:96"/>
( ( <font color="Maroon" title="c3">p1</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> &amp; <font color="Maroon" title="c4">p2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> ) iff  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font> )
 <b>by </b><i><a class="ref" href="oppcat_1.html#D4" title="OPPCAT_1:def.4">OPPCAT_1:def 4</a>, <a class="ref" href="oppcat_1.html#D5" title="OPPCAT_1:def.5">OPPCAT_1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
