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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K17" title="INDEX_1:func.17">-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#K17" title="INDEX_1:func.17">-indexing_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="index_1.html#K13" title="INDEX_1:func.13">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b3">T</font></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K17" title="INDEX_1:func.17">-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#K17" title="INDEX_1:func.17">-indexing_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="index_1.html#K13" title="INDEX_1:func.13">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b2">T</font></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K17" title="INDEX_1:func.17">-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#K17" title="INDEX_1:func.17">-indexing_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="index_1.html#K13" title="INDEX_1:func.13">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">T</font></span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">I</font> = <font color="Maroon" title="c3">F</font> <a href="index_1.html#K17" title="INDEX_1:func.17">-indexing_of</a> <font color="Maroon" title="c1">C</font>;<br/>
<span class="kw">let </span><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#K17" title="INDEX_1:func.17">-indexing_of</a> <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K17" title="INDEX_1:func.17">-indexing_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="index_1.html#K13" title="INDEX_1:func.13">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c5">T</font></span><br/>

<a NAME="E1:59"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="index_1.html#K17" title="INDEX_1:func.17">-indexing_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="index_1.html#K12" title="INDEX_1:func.12">`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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<a NAME="E2:59"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E3:59"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="index_1.html#K17" title="INDEX_1:func.17">-indexing_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="index_1.html#K8" title="INDEX_1:func.8">`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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E28">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K17" title="INDEX_1:func.17">-indexing_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="index_1.html#K13" title="INDEX_1:func.13">-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/><span class="kw">assume </span><a NAME="E1:59_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K17" title="INDEX_1:func.17">-indexing_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="index_1.html#K13" title="INDEX_1:func.13">-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/><span class="kw">then </span><span class="kw">reconsider </span><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/><span class="kw">set </span><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="E3:59_1"/><span class="lab"><font color="Green" title="E29">A5</font></span>: 
(  <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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T105" title="CAT_1:th.105">CAT_1:105</a></span>;<br/><a NAME="E4:59_1"/><span class="lab"><font color="Green" title="E30">A6</font></span>: 
(  <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>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_5.html#T13" title="CAT_5:th.13">CAT_5:13</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E6:59_1"/><span class="lab"><font color="Green" title="E31">A7</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_5.html#D6" title="CAT_5:def.6">CAT_5:def 6</a></span>;<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#K17" title="INDEX_1:func.17">-indexing_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="index_1.html#K12" title="INDEX_1:func.12">`2</a> </span>)</span> <a href="index_1.html#K11" title="INDEX_1:func.11">.</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> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="ref" href="mcart_1.html#D13" title="MCART_1:def.13">MCART_1:def 13</a></span>
<br/>.= 
<font color="Maroon" title="c9">g</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:59_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
;<br/><span class="kw">hence </span><a NAME="E8:59_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#K17" title="INDEX_1:func.17">-indexing_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="index_1.html#K13" title="INDEX_1:func.13">-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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E4:59_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E6:59_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E3:59_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="ref" href="index_1.html#D11" target="_self" title="INDEX_1:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E5:59"/>
 <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#K17" title="INDEX_1:func.17">-indexing_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="index_1.html#K13" title="INDEX_1:func.13">-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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:59"/>
<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#K17" title="INDEX_1:func.17">-indexing_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="index_1.html#K13" title="INDEX_1:func.13">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c5">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
