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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">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">T1</font>, <font color="Olive" title="b3">T2</font> being   <a href="index_1.html#M4" title="INDEX_1:mode.4">TargetCat</a> of <font color="Olive" title="b1">I</font> holds <br/> ( <font color="Olive" title="b1">I</font> <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">T1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">I</font> <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">T2</font> &amp;  <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#K13" title="INDEX_1:func.13">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b2">T1</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <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#K13" title="INDEX_1:func.13">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b3">T2</font></span>)</span> )</span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">T1</font>, <font color="Olive" title="b2">T2</font> being   <a href="index_1.html#M4" title="INDEX_1:mode.4">TargetCat</a> of <font color="Maroon" title="c2">I</font> holds <br/> ( <font color="Maroon" title="c2">I</font> <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">T1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c2">I</font> <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">T2</font> &amp;  <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#K13" title="INDEX_1:func.13">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b1">T1</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <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#K13" title="INDEX_1:func.13">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Olive" title="b2">T2</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">T1</font>, <font color="Maroon" title="c4">T2</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">I</font> <a href="index_1.html#K13" title="INDEX_1:func.13">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">T1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c2">I</font> <a href="index_1.html#K13" title="INDEX_1:func.13">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c4">T2</font> &amp;  <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#K13" title="INDEX_1:func.13">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">T1</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <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#K13" title="INDEX_1:func.13">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c4">T2</font></span>)</span> )</span><br/>





<div><span class="lab"><font color="Green" title="E19">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">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="c5">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 <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K13" title="INDEX_1:func.13">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">T1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">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#K13" title="INDEX_1:func.13">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c4">T2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:47_1"/>
<font color="Maroon" title="c5">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"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K13" title="INDEX_1:func.13">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">T1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">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#K13" title="INDEX_1:func.13">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c4">T2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">f</font> = <font color="Maroon" title="c5">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">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K13" title="INDEX_1:func.13">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">T1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> = 
<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#K8" title="INDEX_1:func.8">`1</a> </span>)</span> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <span class="p4">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">f</font></span>)</span></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#K8" title="INDEX_1:func.8">`1</a> </span>)</span> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <span class="p4">(<span class="default"><a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">f</font></span>)</span></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#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="c6">f</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="index_1.html#D11" target="_self" title="INDEX_1:def.11">Def11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K13" title="INDEX_1:func.13">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c4">T2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>
<span class="kw">by</span> <span class="lab"><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="E2:47"/>
(  <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#K13" title="INDEX_1:func.13">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">T1</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> &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#K13" title="INDEX_1:func.13">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c4">T2</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="E3:47"/>
<font color="Maroon" title="c2">I</font> <a href="index_1.html#K13" title="INDEX_1:func.13">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">T1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c2">I</font> <a href="index_1.html#K13" title="INDEX_1:func.13">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c4">T2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E19">A1</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">  <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#K13" title="INDEX_1:func.13">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">T1</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <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#K13" title="INDEX_1:func.13">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c4">T2</font></span>)</span></span><br/>

<div><span class="lab"><font color="Green" title="E20">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">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="c5">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#K13" title="INDEX_1:func.13">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">T1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K13" title="INDEX_1:func.13">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c4">T2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:47_2"/>
<font color="Maroon" title="c5">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K13" title="INDEX_1:func.13">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">T1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K13" title="INDEX_1:func.13">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c4">T2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">a</font> = <font color="Maroon" title="c5">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/><span class="kw">thus </span><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#K13" title="INDEX_1:func.13">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">T1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="index_1.html#K8" title="INDEX_1:func.8">`1</a> </span>)</span> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <font color="Maroon" title="c6">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="index_1.html#E23"><span class="lab"><font color="Green" title="E17">Lm2</font></span></a></span>
<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#K13" title="INDEX_1:func.13">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c4">T2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="index_1.html#E23"><span class="lab"><font color="Green" title="E17">Lm2</font></span></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="E5:47"/>
(  <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#K13" title="INDEX_1:func.13">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">T1</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"><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#K13" title="INDEX_1:func.13">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c4">T2</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> )
 
<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="E6:47"/>
 <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#K13" title="INDEX_1:func.13">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">T1</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <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#K13" title="INDEX_1:func.13">-functor</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c4">T2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47"><span class="lab"><font color="Green" title="E20">A2</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/>


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