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<b>let </b><font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c2">B</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">S</font> being   <a href="oppcat_1.html#M1" title="OPPCAT_1:mode.1">Contravariant_Functor</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">B</font><br/> for <font color="Olive" title="b2">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> holds <br/> ( <span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">S</font> <a href="oppcat_1.html#K9" title="OPPCAT_1:func.9">*'</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">S</font> <a href="oppcat_1.html#K9" title="OPPCAT_1:func.9">*'</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">S</font> <a href="oppcat_1.html#K9" title="OPPCAT_1:func.9">*'</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Olive" title="b2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">S</font> <a href="oppcat_1.html#K9" title="OPPCAT_1:func.9">*'</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">f</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c3">S</font> be    <a href="oppcat_1.html#M1" title="OPPCAT_1:mode.1">Contravariant_Functor</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">B</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> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> holds <br/> ( <span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="oppcat_1.html#K9" title="OPPCAT_1:func.9">*'</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="oppcat_1.html#K9" title="OPPCAT_1:func.9">*'</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="oppcat_1.html#K9" title="OPPCAT_1:func.9">*'</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="oppcat_1.html#K9" title="OPPCAT_1:func.9">*'</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="oppcat_1.html#K9" title="OPPCAT_1:func.9">*'</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="oppcat_1.html#K9" title="OPPCAT_1:func.9">*'</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="oppcat_1.html#K9" title="OPPCAT_1:func.9">*'</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="oppcat_1.html#K9" title="OPPCAT_1:func.9">*'</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">f</font></span>)</span> )</span><br/>







<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>now </b></a><div class="add"><b>thus </b><span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="oppcat_1.html#K9" title="OPPCAT_1:func.9">*'</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> 
<b>by </b><i><a class="ref" href="oppcat_1.html#T46" target="_self" title="OPPCAT_1:th.46">Th46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> 
<b>by </b><i><a class="ref" href="oppcat_1.html#T33" target="_self" title="OPPCAT_1:th.33">Th33</a></i>
<br/>.= 
 <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </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"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="oppcat_1.html#K9" title="OPPCAT_1:func.9">*'</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span></span><br/><b>thus </b><span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="oppcat_1.html#K9" title="OPPCAT_1:func.9">*'</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">f</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <font color="Maroon" title="c3">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> 
<b>by </b><i><a class="ref" href="oppcat_1.html#T46" target="_self" title="OPPCAT_1:th.46">Th46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> 
<b>by </b><i><a class="ref" href="oppcat_1.html#T33" target="_self" title="OPPCAT_1:th.33">Th33</a></i>
<br/>.= 
 <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span>

; <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="E2:71"/>
( <span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="oppcat_1.html#K9" title="OPPCAT_1:func.9">*'</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="oppcat_1.html#K9" title="OPPCAT_1:func.9">*'</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cat_1.html#K12" title="CAT_1:func.12">Obj</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="oppcat_1.html#K9" title="OPPCAT_1:func.9">*'</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">S</font> <a href="oppcat_1.html#K9" title="OPPCAT_1:func.9">*'</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">f</font></span>)</span> )
 <b>by </b><i><a class="ref" href="oppcat_1.html#D9" target="_self" title="OPPCAT_1:def.9">Def9</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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