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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">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  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b1">f</font> holds <br/> <a href="cat_5.html#K23" title="CAT_5:func.23">SliceContraFunctor</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_5.html#K23" title="CAT_5:func.23">SliceContraFunctor</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="cat_5.html#K23" title="CAT_5:func.23">SliceContraFunctor</a> <font color="Olive" title="b2">g</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">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"> (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c2">f</font> implies  <a href="cat_5.html#K23" title="CAT_5:func.23">SliceContraFunctor</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_5.html#K23" title="CAT_5:func.23">SliceContraFunctor</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="cat_5.html#K23" title="CAT_5:func.23">SliceContraFunctor</a> <font color="Maroon" title="c3">g</font></span>)</span> )</span><br/>



<b>assume </b><a NAME="E1:62"/><i><font color="Green" title="E40">A1</font></i>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c2">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="cat_5.html#K23" title="CAT_5:func.23">SliceContraFunctor</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_5.html#K23" title="CAT_5:func.23">SliceContraFunctor</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="cat_5.html#K23" title="CAT_5:func.23">SliceContraFunctor</a> <font color="Maroon" title="c3">g</font></span>)</span></span><br/>

<a NAME="E2:62"/><b>then </b><i><font color="Green" title="E41">A2</font></i>: 
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c3">g</font> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T42" title="CAT_1:th.42">CAT_1:42</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">A1</font> = <span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</font>;<br/>
<b>set </b><font color="Maroon" title="c5">A2</font> = <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</font>;<br/>
<b>set </b><font color="Maroon" title="c6">A3</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">g</font></span>)</span> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">F</font> =  <a href="cat_5.html#K23" title="CAT_5:func.23">SliceContraFunctor</a> <font color="Maroon" title="c2">f</font> as    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</font>,<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">G</font> =  <a href="cat_5.html#K23" title="CAT_5:func.23">SliceContraFunctor</a> <font color="Maroon" title="c3">g</font> as    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</font>,<span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="txt" href="#E1:62"><i><font color="Green" title="E40">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">FG</font> =  <a href="cat_5.html#K23" title="CAT_5:func.23">SliceContraFunctor</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">f</font></span>)</span> as    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</font>,<span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="txt" href="#E2:62"><i><font color="Green" title="E41">A2</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">m</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <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">g</font></span>)</span> <a href="cat_5.html#K15" title="CAT_5:func.15">-SliceCat</a> <font color="Maroon" title="c1">C</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c8">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">FG</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m</font></span><br/><a NAME="E1:62_1"/>
<font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><span class="p2"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</a> </span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>
 <b>by </b><i><a class="ref" href="cat_5.html#D14" target="_self" title="CAT_5:def.14">Def14</a></i>;<br/><a NAME="E2:62_1"/><b>then </b><i><font color="Green" title="E42">A3</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">g</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">g</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">m</font> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</a>  )
 <b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a>, <a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</a></i>;<br/><a NAME="E3:62_1"/>
(  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> </span>)</span> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="cat_5.html#T24" target="_self" title="CAT_5:th.24">Th24</a></i>;<br/><a NAME="E4:62_1"/><b>then </b><i><font color="Green" title="E43">A4</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">f</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">f</font> )
 <b>by </b><i><a class="txt" href="#E1:62"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="cat_1.html#T43" title="CAT_1:th.43">CAT_1:43</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c8">G</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m</font> = 
<font color="Maroon" title="c7">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><span class="p2"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="cat_5.html#K20" title="CAT_5:func.20">`11</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="cat_5.html#K21" title="CAT_5:func.21">`12</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">f</font></span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="cat_5.html#K19" title="CAT_5:func.19">`2</a> </span>)</span></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>
<b>by </b><i><a class="txt" href="#E2:62_1"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="cat_5.html#D14" target="_self" title="CAT_5:def.14">Def14</a></i>
<br/>.= 
<font color="Maroon" title="c9">FG</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m</font>
<b>by </b><i><a class="txt" href="#E2:62"><i><font color="Green" title="E41">A2</font></i></a>, <a class="txt" href="#E4:62_1"><i><font color="Green" title="E43">A4</font></i></a>, <a class="ref" href="cat_5.html#D14" target="_self" title="CAT_5:def.14">Def14</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>hence </b><a NAME="E7:62"/>
 <a href="cat_5.html#K23" title="CAT_5:func.23">SliceContraFunctor</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_5.html#K23" title="CAT_5:func.23">SliceContraFunctor</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="cat_5.html#K23" title="CAT_5:func.23">SliceContraFunctor</a> <font color="Maroon" title="c3">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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