<?xml version="1.0"?>
<div class="add">

<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> holds  <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Olive" title="b2">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Olive" title="b2">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is   <a href="index_1.html#NM9" title="INDEX_1:NM.9">Indexing</a> of <font color="Maroon" title="c1">C</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> holds  <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Olive" title="b1">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Olive" title="b1">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is   <a href="index_1.html#NM9" title="INDEX_1:NM.9">Indexing</a> of <font color="Maroon" title="c1">C</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"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<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>,<span class="p2">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c3">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is   <a href="index_1.html#NM9" title="INDEX_1:NM.9">Indexing</a> of <font color="Maroon" title="c1">C</font></span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">I</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<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>,<span class="p2">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c3">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<a NAME="E1:56"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<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>,<span class="p2">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c3">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`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/>
<a NAME="E2:56"/><span class="lab"><font color="Green" title="E25">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="E3:56"/>
 <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>
 
<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="E4:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A3</font></span>: 
 <a href="cat_1.html#K8" title="CAT_1:func.8">Obj</a> <font color="Maroon" title="c3">F</font> is   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of 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="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<a NAME="E5:56"/><span class="lab"><font color="Green" title="E27">A4</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<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>,<span class="p2">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c3">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`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="E6:56"/><span class="lab"><font color="Green" title="E28">A5</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#D13" title="MCART_1:def.13">MCART_1:def 13</a></span>;<br/>
<a NAME="E7:56"/><span class="kw">then </span>
 <a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c3">F</font> is   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of 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="txt" href="#E2:56"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a>, <a class="ref" href="relat_1.html#D18" title="RELAT_1:def.18">RELAT_1:def 18</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">I</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<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>,<span class="p2">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c3">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="index_1.html#M1" title="INDEX_1:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>,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="txt" href="#E4:56"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="index_1.html#D4" target="_self" title="INDEX_1:def.4">Def4</a></span>;<br/>
<a NAME="E9:56"/>
 <a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c3">F</font> is  <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><span class="kw">proof </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> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="funcop_1.html#D6" title="FUNCOP_1:def.6">FUNCOP_1:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c3">F</font></span>)</span> or <span class="p1">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> is    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:56_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c3">F</font></span>)</span>
 ; <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="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> is    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">x</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="mcart_1.html#D13" title="MCART_1:def.13">MCART_1:def 13</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">m</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">x</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">D</font> ;<br/>
<a NAME="E4:56_1"/>
<span class="p1">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">m</font> <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="#E2:56"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="mcart_1.html#D13" title="MCART_1:def.13">MCART_1:def 13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:56_1"/>
<span class="p1">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> is    <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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">FF</font> =  <a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c3">F</font> as   <a href="pboole.html#NM2" title="PBOOLE:NM.2">ManySortedFunction</a> of 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="txt" href="#E6:56"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E2:56"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a>, <a class="ref" href="relat_1.html#D18" title="RELAT_1:def.18">RELAT_1:def 18</a></span>;<br/>
<a NAME="E11:56"/>
<font color="Maroon" title="c5">I</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  is  <a href="index_1.html#V1" title="INDEX_1:attr.1">Category-yielding</a> 
 
<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/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">I</font> = <font color="Maroon" title="c5">I</font> as   <a href="index_1.html#V2" title="INDEX_1:attr.2">Category-yielding_on_first</a>   <a href="index_1.html#M1" title="INDEX_1:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>,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="index_1.html#D5" target="_self" title="INDEX_1:def.5">Def5</a></span>;<br/>
<a NAME="E13:56"/>
<font color="Maroon" title="c6">FF</font> is    <a href="index_1.html#M2" title="INDEX_1:mode.2">ManySortedFunctor</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="index_1.html#K8" title="INDEX_1:func.8">`1</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c1">C</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="index_1.html#K8" title="INDEX_1:func.8">`1</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c1">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">m</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="index_1.html#D7" target="_self" title="INDEX_1:def.7">INDEX_1:def 7</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">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font> is    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="index_1.html#K8" title="INDEX_1:func.8">`1</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c1">C</font></span>)</span> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <font color="Maroon" title="c8">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="index_1.html#K8" title="INDEX_1:func.8">`1</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c1">C</font></span>)</span> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">m</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">D</font> ;<br/>
<a NAME="E2:56_2"/><span class="lab"><font color="Green" title="E29">A6</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="cat_5.html#K8" title="CAT_5:func.8">`11</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c8">m</font></span>)</span>
 
<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/>
<a NAME="E3:56_2"/>
<font color="Maroon" title="c9">x</font> <a href="cat_5.html#K9" title="CAT_5:func.9">`12</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c8">m</font></span>)</span>
 
<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="c10">f</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c9">x</font> <a href="cat_5.html#K8" title="CAT_5:func.8">`11</a> ,<font color="Maroon" title="c9">x</font> <a href="cat_5.html#K9" title="CAT_5:func.9">`12</a> <span class="kw"> such that </span><br/><a NAME="E5:56_2"/><span class="lab"><font color="Green" title="E30">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="c8">m</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"><font color="Maroon" title="c9">x</font> <a href="cat_5.html#K8" title="CAT_5:func.8">`11</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">x</font> <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="c10">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_2"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="ref" href="cat_5.html#D6" title="CAT_5:def.6">CAT_5:def 6</a></span>;<br/>
<span class="lab"><font color="Green" title="E31">A8</font></span>: <span class="p1">(<span class="default"><span class="p2">(<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="partfun1.html#K1" title="PARTFUN1:func.1">*</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font> = 
<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="c8">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>
<br/>.= 
 <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="c8">m</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/>
<span class="lab"><font color="Green" title="E32">A9</font></span>: <span class="p1">(<span class="default"><span class="p2">(<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="partfun1.html#K1" title="PARTFUN1:func.1">*</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font> = 
<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="c8">m</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>
<br/>.= 
 <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="c8">m</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/>
<font color="Maroon" title="c6">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font> = 
<font color="Maroon" title="c9">x</font> <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="#E2:56"><span class="lab"><font color="Green" title="E25">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="c10">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56_2"><span class="lab"><font color="Green" title="E30">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="E9:56_2"/>
<font color="Maroon" title="c6">FF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font> is    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="index_1.html#K8" title="INDEX_1:func.8">`1</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c1">C</font></span>)</span> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <font color="Maroon" title="c8">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="index_1.html#K8" title="INDEX_1:func.8">`1</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c1">C</font></span>)</span> <a href="index_1.html#K2" title="INDEX_1:func.2">.</a> <font color="Maroon" title="c8">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E2:56_2"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="txt" href="#E6:56_2"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="txt" href="#E7:56_2"><span class="lab"><font color="Green" title="E32">A9</font></span></a>, <a class="ref" href="cat_5.html#T13" title="CAT_5:th.13">CAT_5:13</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>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">I</font> = <font color="Maroon" title="c7">I</font> as    <a href="index_1.html#M3" title="INDEX_1:mode.3">Indexing</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c1">C</font>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="ref" href="index_1.html#D8" target="_self" title="INDEX_1:def.8">Def8</a></span>;<br/>
<a NAME="E15:56"/><span class="lab"><font color="Green" title="E29">A10</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/>
<div><span class="lab"><font color="Green" title="E30">A11</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">m1</font>, <font color="Maroon" title="c10">m2</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c10">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c9">m1</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c8">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> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c9">m1</font></span>)</span> <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="c8">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="c10">m2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">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="c9">m1</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:56_3"/><span class="lab"><font color="Green" title="E31">A12</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c10">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c9">m1</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="c8">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> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c9">m1</font></span>)</span> <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="c8">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="c10">m2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">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="c9">m1</font></span>)</span></span><br/><span class="kw">set </span><font color="Maroon" title="c11">h1</font> = <font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">m1</font>;<br/><span class="kw">set </span><font color="Maroon" title="c12">h2</font> = <font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">m2</font>;<br/><span class="lab"><font color="Green" title="E32">A13</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="c10">m2</font></span>)</span> = 
<font color="Maroon" title="c3">F</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p1">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c10">m2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T109" title="CAT_1:th.109">CAT_1:109</a></span>
<br/>.= 
 <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="c9">m1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_3"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="ref" href="cat_1.html#T109" title="CAT_1:th.109">CAT_1:109</a></span>
;<br/><a NAME="E3:56_3"/><span class="lab"><font color="Green" title="E33">A14</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="c10">m2</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="c10">m2</font></span>)</span> <a href="cat_5.html#K8" title="CAT_5:func.8">`11</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/><a NAME="E4:56_3"/>
 <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="c10">m2</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="c10">m2</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="c13">f2</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="c10">m2</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="c10">m2</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="E6:56_3"/><span class="lab"><font color="Green" title="E34">A15</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="c10">m2</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="c10">m2</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="c10">m2</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="c13">f2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_3"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="ref" href="cat_5.html#D6" title="CAT_5:def.6">CAT_5:def 6</a></span>;<br/><a NAME="E7:56_3"/><span class="lab"><font color="Green" title="E35">A16</font></span>: 
 <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="c9">m1</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="c9">m1</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/><a NAME="E8:56_3"/>
 <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="c9">m1</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="c9">m1</font></span>)</span> <a href="cat_5.html#K8" title="CAT_5:func.8">`11</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="c14">f1</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="c9">m1</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="c9">m1</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="E10:56_3"/><span class="lab"><font color="Green" title="E36">A17</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="c9">m1</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="c9">m1</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="c9">m1</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="c14">f1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_3"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="ref" href="cat_5.html#D6" title="CAT_5:def.6">CAT_5:def 6</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c8">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> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c9">m1</font></span>)</span> = 
<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> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c9">m1</font></span>)</span></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="#E5:56"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E15:56"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="ref" href="mcart_1.html#D13" title="MCART_1:def.13">MCART_1:def 13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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="c10">m2</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p2">(<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="c9">m1</font></span>)</span></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="#E1:56_3"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="ref" href="cat_1.html#T99" title="CAT_1:th.99">CAT_1:99</a></span>
<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="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">m1</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="c10">m2</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>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c14">f1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_3"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="txt" href="#E7:56_3"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="txt" href="#E6:56_3"><span class="lab"><font color="Green" title="E34">A15</font></span></a>, <a class="txt" href="#E10:56_3"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="txt" href="#E2:56_3"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="ref" href="cat_5.html#D6" title="CAT_5:def.6">CAT_5:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c13">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c14">f1</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/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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="c10">m2</font></span>)</span> <a href="index_1.html#K1" title="INDEX_1:func.1">`2</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c14">f1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:56_3"><span class="lab"><font color="Green" title="E34">A15</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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="c10">m2</font></span>)</span> <a href="index_1.html#K1" title="INDEX_1:func.1">`2</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<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="c9">m1</font></span>)</span> <a href="index_1.html#K1" title="INDEX_1:func.1">`2</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:56_3"><span class="lab"><font color="Green" title="E36">A17</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">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="c10">m2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<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="c9">m1</font></span>)</span> <a href="index_1.html#K1" title="INDEX_1:func.1">`2</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E15:56"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="ref" href="mcart_1.html#D13" title="MCART_1:def.13">MCART_1:def 13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">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="c10">m2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">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="c9">m1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E15:56"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="ref" href="mcart_1.html#D13" title="MCART_1:def.13">MCART_1:def 13</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</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="c8">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> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">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="c9">a</font></span>)</span></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c10">i</font> = <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> <font color="Maroon" title="c9">a</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">D</font> ;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c8">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> <span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c9">a</font></span>)</span> = 
<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> <span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c9">a</font></span>)</span></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="#E5:56"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E15:56"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="ref" href="mcart_1.html#D13" title="MCART_1:def.13">MCART_1:def 13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">id</a> <font color="Maroon" title="c10">i</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="ref" href="cat_1.html#T104" title="CAT_1:th.104">CAT_1:104</a></span>
<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="c8">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="c9">a</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">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="c9">a</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">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="c9">a</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="cat_5.html#D6" title="CAT_5:def.6">CAT_5:def 6</a></span>
<br/>.= 
 <a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">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="c9">a</font></span>)</span>
<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>
; <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>
<span class="kw">hence </span><a NAME="E18:56"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<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>,<span class="p2">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <font color="Maroon" title="c3">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is   <a href="index_1.html#NM9" title="INDEX_1:NM.9">Indexing</a> of <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:56"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="index_1.html#T6" target="_self" title="INDEX_1:th.6">Th6</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>
