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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">I</font> being  <a href="index_1.html#NM9" title="INDEX_1:NM.9">Indexing</a> of <font color="Maroon" title="c1">C</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">I</font> holds   <a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">I</font> <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>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">I</font> <a href="index_1.html#K10" title="INDEX_1:func.10">`1</a> </span><br/><b>let </b><font color="Maroon" title="c2">I</font> be   <a href="index_1.html#NM9" title="INDEX_1:NM.9">Indexing</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">T</font> being   <a href="index_1.html#M4" title="INDEX_1:mode.4">TargetCat</a> of <font color="Maroon" title="c2">I</font> holds   <a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <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>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">I</font> <a href="index_1.html#K10" title="INDEX_1:func.10">`1</a> </span><br/><b>let </b><font color="Maroon" title="c3">T</font> be    <a href="index_1.html#M4" title="INDEX_1:mode.4">TargetCat</a> of <font color="Maroon" title="c2">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K15" title="INDEX_1:func.15">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">I</font> <a href="index_1.html#K10" title="INDEX_1:func.10">`1</a> </span><br/>





<a NAME="E1:44"/><i><font color="Green" title="E18">A1</font></i>: 
(  <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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K15" title="INDEX_1:func.15">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">T</font></span>)</span></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> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K10" title="INDEX_1:func.10">`1</a> </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>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> implies <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K15" title="INDEX_1:func.15">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K10" title="INDEX_1:func.10">`1</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> )</span><br/><b>assume </b><a NAME="E1:44_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">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"> <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K15" title="INDEX_1:func.15">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K10" title="INDEX_1:func.10">`1</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span><br/><b>then reconsider </b><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c4">x</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font> ;<br/><a NAME="E3:44_1"/><i><font color="Green" title="E19">A2</font></i>: 
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> )
 <b>by </b><i><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></i>;<br/><span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K15" title="INDEX_1:func.15">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">T</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font> = 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K15" title="INDEX_1:func.15">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">T</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></i>
<br/>.= 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K15" title="INDEX_1:func.15">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">T</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#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T104" title="CAT_1:th.104">CAT_1:104</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K15" title="INDEX_1:func.15">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="mcart_1.html#K17" title="MCART_1:func.17">`11</a> 
<b>by </b><i><a class="ref" href="cat_5.html#T2" title="CAT_5:th.2">CAT_5:2</a></i>
<br/>.= 
<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="c2">I</font> <a href="index_1.html#K10" title="INDEX_1:func.10">`1</a> </span>)</span> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K10" title="INDEX_1:func.10">`1</a> </span>)</span> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <font color="Maroon" title="c5">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">I</font> <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> <span class="p3">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K17" title="MCART_1:func.17">`11</a> 
<b>by </b><i><a class="txt" href="#E3:44_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="index_1.html#D11" target="_self" title="INDEX_1:def.11">Def11</a></i>
;<br/><b>hence </b><a NAME="E5:44_1"/>
<span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K15" title="INDEX_1:func.15">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K10" title="INDEX_1:func.10">`1</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="ref" href="mcart_1.html#T89" title="MCART_1:th.89">MCART_1:89</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="E3:44"/>
 <a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K15" title="INDEX_1:func.15">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">I</font> <a href="index_1.html#K10" title="INDEX_1:func.10">`1</a> 
 <b>by </b><i><a class="txt" href="#E1:44"><i><font color="Green" title="E18">A1</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/>


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