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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">D</font> being <a href="cat_5.html#V3" title="CAT_5:attr.3">Categorial</a> <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a><br/> for <font color="Olive" title="b2">F</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">D</font><br/> for <font color="Olive" title="b3">T</font> being   <a href="index_1.html#M4" title="INDEX_1:mode.4">TargetCat</a> of <font color="Olive" title="b2">F</font> <a href="index_1.html#K19" title="INDEX_1:func.19">-indexing_of</a> <font color="Maroon" title="c1">C</font> holds  <font color="Olive" title="b2">F</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="index_1.html#K19" title="INDEX_1:func.19">-indexing_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="index_1.html#K15" title="INDEX_1:func.15">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b3">T</font></span><br/><b>let </b><font color="Maroon" title="c2">D</font> be  <a href="cat_5.html#V3" title="CAT_5:attr.3">Categorial</a> <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> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">D</font><br/> for <font color="Olive" title="b2">T</font> being   <a href="index_1.html#M4" title="INDEX_1:mode.4">TargetCat</a> of <font color="Olive" title="b1">F</font> <a href="index_1.html#K19" title="INDEX_1:func.19">-indexing_of</a> <font color="Maroon" title="c1">C</font> holds  <font color="Olive" title="b1">F</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="index_1.html#K19" title="INDEX_1:func.19">-indexing_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="index_1.html#K15" title="INDEX_1:func.15">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b2">T</font></span><br/><b>let </b><font color="Maroon" title="c3">F</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">D</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">T</font> being   <a href="index_1.html#M4" title="INDEX_1:mode.4">TargetCat</a> of <font color="Maroon" title="c3">F</font> <a href="index_1.html#K19" title="INDEX_1:func.19">-indexing_of</a> <font color="Maroon" title="c1">C</font> holds  <font color="Maroon" title="c3">F</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="index_1.html#K19" title="INDEX_1:func.19">-indexing_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="index_1.html#K15" title="INDEX_1:func.15">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">T</font></span><br/>





<b>set </b><font color="Maroon" title="c4">I</font> = <font color="Maroon" title="c3">F</font> <a href="index_1.html#K19" title="INDEX_1:func.19">-indexing_of</a> <font color="Maroon" title="c1">C</font>;<br/>
<b>let </b><font color="Maroon" title="c5">T</font> be    <a href="index_1.html#M4" title="INDEX_1:mode.4">TargetCat</a> of <font color="Maroon" title="c3">F</font> <a href="index_1.html#K19" title="INDEX_1:func.19">-indexing_of</a> <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="c3">F</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="index_1.html#K19" title="INDEX_1:func.19">-indexing_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="index_1.html#K15" title="INDEX_1:func.15">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c5">T</font></span><br/>

<a NAME="E1:57"/><i><font color="Green" title="E25">A1</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="index_1.html#K19" title="INDEX_1:func.19">-indexing_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="index_1.html#K10" title="INDEX_1:func.10">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Maroon" title="c3">F</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="index_1.html#K19" title="INDEX_1:func.19">-indexing_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="index_1.html#K14" title="INDEX_1:func.14">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c3">F</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E2:57"/><i><font color="Green" title="E26">A2</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:57"/><i><font color="Green" title="E27">A3</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="index_1.html#K19" title="INDEX_1:func.19">-indexing_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="index_1.html#K15" title="INDEX_1:func.15">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c5">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <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="c3">F</font> <a href="index_1.html#K19" title="INDEX_1:func.19">-indexing_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="index_1.html#K15" title="INDEX_1:func.15">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c5">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> )</span><br/><b>assume </b><a NAME="E1:57_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</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="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <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="c3">F</font> <a href="index_1.html#K19" title="INDEX_1:func.19">-indexing_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="index_1.html#K15" title="INDEX_1:func.15">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c5">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font></span><br/><b>then reconsider </b><font color="Maroon" title="c7">f</font> = <font color="Maroon" title="c6">x</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font> ;<br/><b>set </b><font color="Maroon" title="c8">h</font> = <font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f</font>;<br/><a NAME="E3:57_1"/><i><font color="Green" title="E28">A4</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">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="cat_5.html#K8" title="CAT_5:func.8">`11</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="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="cat_5.html#K9" title="CAT_5:func.9">`12</a>  )
 <b>by </b><i><a class="ref" href="cat_5.html#T13" title="CAT_5:th.13">CAT_5:13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">g</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="cat_5.html#K8" title="CAT_5:func.8">`11</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="cat_5.html#K9" title="CAT_5:func.9">`12</a> <b> such that </b><br/><a NAME="E5:57_1"/><i><font color="Green" title="E29">A5</font></i>: 
<font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="cat_5.html#K8" title="CAT_5:func.8">`11</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="cat_5.html#K9" title="CAT_5:func.9">`12</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c9">g</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="cat_5.html#D6" title="CAT_5:def.6">CAT_5:def 6</a></i>;<br/><a NAME="E6:57_1"/><i><font color="Green" title="E30">A6</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">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c7">f</font></span>)</span> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c7">f</font></span>)</span> )
 <b>by </b><i><a class="ref" href="cat_1.html#T105" title="CAT_1:th.105">CAT_1:105</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="index_1.html#K19" title="INDEX_1:func.19">-indexing_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="index_1.html#K14" title="INDEX_1:func.14">`2</a> </span>)</span> <a href="index_1.html#K13" title="INDEX_1:func.13">.</a> <font color="Maroon" title="c7">f</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="index_1.html#K1" title="INDEX_1:func.1">`2</a> 
<b>by </b><i><a class="txt" href="#E1:57"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="#E2:57"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="mcart_1.html#D13" title="MCART_1:def.13">MCART_1:def 13</a></i>
<br/>.= 
<font color="Maroon" title="c9">g</font>
<b>by </b><i><a class="txt" href="#E5:57_1"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/><b>hence </b><a NAME="E8:57_1"/>
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <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="c3">F</font> <a href="index_1.html#K19" title="INDEX_1:func.19">-indexing_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="index_1.html#K15" title="INDEX_1:func.15">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c5">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="#E1:57"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="#E3:57_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="#E5:57_1"><i><font color="Green" title="E29">A5</font></i></a>, <a class="txt" href="#E6:57_1"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="index_1.html#D11" target="_self" title="INDEX_1:def.11">Def11</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="E5:57"/>
<font color="Maroon" title="c3">F</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="index_1.html#K19" title="INDEX_1:func.19">-indexing_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="index_1.html#K15" title="INDEX_1:func.15">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c5">T</font>
 <b>by </b><i><a class="txt" href="#E2:57"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="#E3:57"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</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>
