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<b>let </b><font color="Maroon" title="c1">A</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/><b>let </b><font color="Maroon" title="c2">O</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">A</font>;<br/><b>let </b><font color="Maroon" title="c3">M</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">A</font>;<br/><b>let </b><font color="Maroon" title="c4">DOM</font>, <font color="Maroon" title="c5">COD</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">O</font>;<br/>







<b>assume </b><a NAME="E1:11"/><i><font color="Green" title="E10">A1</font></i>: 
( <font color="Maroon" title="c4">DOM</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">A</font> <a href="nattra_1.html#K1" title="NATTRA_1:func.1">|</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c5">COD</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">A</font> <a href="nattra_1.html#K1" title="NATTRA_1:func.1">|</a> <font color="Maroon" title="c3">M</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c6">COMP</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c3">M</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c3">M</font>;<br/>

<b>assume </b><a NAME="E2:11"/><i><font color="Green" title="E11">A2</font></i>: 
<font color="Maroon" title="c6">COMP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">A</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">M</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c7">ID</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>;<br/>

<b>assume </b><a NAME="E3:11"/><i><font color="Green" title="E12">A3</font></i>: 
<font color="Maroon" title="c7">ID</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">A</font> <a href="nattra_1.html#K1" title="NATTRA_1:func.1">|</a> <font color="Maroon" title="c2">O</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c8">C</font> =  <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #);<br/>
<a NAME="E4:11"/><i><font color="Green" title="E13">A4</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E2:11"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></i>;<br/>
<div><i><font color="Green" title="E14">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">f</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">A</font>;<br/><b>let </b><font color="Maroon" title="c10">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #);<br/><b>assume </b><a NAME="E1:11_1"/><i><font color="Green" title="E15">A6</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font>
 ;<br/><a NAME="E2:11_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #)
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E3:11_1"/>
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E1:11"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="nattra_1.html#E1:11_1"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><a NAME="E4:11_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #)
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E5:11_1"/>
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E1:11"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="nattra_1.html#E1:11_1"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:11"/><i><font color="Green" title="E15">A7</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">A</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #),the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #)</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E2:11"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/>
<div><i><font color="Green" title="E16">A8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">f</font>, <font color="Maroon" title="c10">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #);<br/><b>reconsider </b><font color="Maroon" title="c11">g'</font> = <font color="Maroon" title="c10">g</font>, <font color="Maroon" title="c12">f'</font> = <font color="Maroon" title="c9">f</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>assume </b><a NAME="E2:11_2"/>
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f</font>
 ;<br/><b>then </b>the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g'</font> = 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E5:11"><i><font color="Green" title="E14">A5</font></i></a></i>
<br/>.= 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f'</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E5:11"><i><font color="Green" title="E14">A5</font></i></a></i>
;<br/><a NAME="E4:11_2"/><b>then </b><i><font color="Green" title="E17">A9</font></i>: 
<span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c11">g'</font>,<font color="Maroon" title="c12">f'</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a></i>;<br/><a NAME="E5:11_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #),the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #)</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E6:11_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #)
 <b>by </b><i><a class="txt" href="nattra_1.html#E6:11"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="nattra_1.html#E4:11_2"><i><font color="Green" title="E17">A9</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:11"/>
 <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) is <a href="cat_1.html#V3" title="CAT_1:attr.3">Category-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:11_3"/>
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) iff the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> )
  <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">f</font>, <font color="Maroon" title="c10">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #);<br/>

<b>reconsider </b><font color="Maroon" title="c11">g'</font> = <font color="Maroon" title="c10">g</font>, <font color="Maroon" title="c12">f'</font> = <font color="Maroon" title="c9">f</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>thus </b><a NAME="E2:11_3_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) implies the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_3_1_1"/><i><font color="Green" title="E17">A10</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #)
 ;<br/>

<b>thus </b>the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font> = 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g'</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E5:11"><i><font color="Green" title="E14">A5</font></i></a></i>
<br/>.= 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f'</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E4:11"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="nattra_1.html#E1:11_3_1_1"><i><font color="Green" title="E17">A10</font></i></a>, <a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a></i>
<br/>.= 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E5:11"><i><font color="Green" title="E14">A5</font></i></a></i>
;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:11_3_1"/>
( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) )
 <b>by </b><i><a class="txt" href="nattra_1.html#E7:11"><i><font color="Green" title="E16">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:11_3"/>
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #)  st the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> holds <br/>( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">f</font>, <font color="Maroon" title="c10">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #);<br/>

<b>reconsider </b><font color="Maroon" title="c11">g'</font> = <font color="Maroon" title="c10">g</font>, <font color="Maroon" title="c12">f'</font> = <font color="Maroon" title="c9">f</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>assume </b><a NAME="E2:11_3_2"/><i><font color="Green" title="E17">A11</font></i>: 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f</font>
 ;<br/>

<b>then </b><i><font color="Green" title="E18">A12</font></i>: the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g'</font> = 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E5:11"><i><font color="Green" title="E14">A5</font></i></a></i>
<br/>.= 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f'</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E5:11"><i><font color="Green" title="E14">A5</font></i></a></i>
;<br/>
<a NAME="E4:11_3_2"/><i><font color="Green" title="E19">A13</font></i>: 
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c11">g'</font>,<font color="Maroon" title="c12">f'</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E2:11"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="nattra_1.html#E7:11"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="nattra_1.html#E2:11_3_2"><i><font color="Green" title="E17">A11</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<a NAME="E5:11_3_2"/><i><font color="Green" title="E20">A14</font></i>: 
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #)
 
<b>by </b><i><a class="txt" href="nattra_1.html#E7:11"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="nattra_1.html#E2:11_3_2"><i><font color="Green" title="E17">A11</font></i></a>, <a class="ref" href="partfun1.html#T27" title="PARTFUN1:th.27">PARTFUN1:27</a></i>;<br/>
<a NAME="E6:11_3_2"/><b>then </b><i><font color="Green" title="E21">A15</font></i>: 
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c11">g'</font>,<font color="Maroon" title="c12">f'</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E4:11_3_2"><i><font color="Green" title="E19">A13</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>hence </b>the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font></span>)</span> = 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">g'</font>,<font color="Maroon" title="c12">f'</font></span>)</span>
<b>by </b><i><a class="txt" href="nattra_1.html#E5:11"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="nattra_1.html#E4:11_3_2"><i><font color="Green" title="E19">A13</font></i></a>, <a class="txt" href="nattra_1.html#E5:11_3_2"><i><font color="Green" title="E20">A14</font></i></a></i>
<br/>.= 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f'</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E3:11_3_2"><i><font color="Green" title="E18">A12</font></i></a>, <a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a></i>
<br/>.= 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E5:11"><i><font color="Green" title="E14">A5</font></i></a></i>
;<br/><br/>

<b>thus </b>the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font></span>)</span> = 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">g'</font>,<font color="Maroon" title="c12">f'</font></span>)</span>
<b>by </b><i><a class="txt" href="nattra_1.html#E5:11"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="nattra_1.html#E4:11_3_2"><i><font color="Green" title="E19">A13</font></i></a>, <a class="txt" href="nattra_1.html#E5:11_3_2"><i><font color="Green" title="E20">A14</font></i></a>, <a class="txt" href="nattra_1.html#E6:11_3_2"><i><font color="Green" title="E21">A15</font></i></a></i>
<br/>.= 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g'</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E3:11_3_2"><i><font color="Green" title="E18">A12</font></i></a>, <a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a></i>
<br/>.= 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E5:11"><i><font color="Green" title="E14">A5</font></i></a></i>
;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:11_3"/>
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #)  st the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> &amp; the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> holds <br/>the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b3">h</font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b3">h</font>,<font color="Olive" title="b2">g</font></span>)</span>,<font color="Olive" title="b1">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">f</font>, <font color="Maroon" title="c10">g</font>, <font color="Maroon" title="c11">h</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #);<br/>

<b>reconsider </b><font color="Maroon" title="c12">g'</font> = <font color="Maroon" title="c10">g</font>, <font color="Maroon" title="c13">f'</font> = <font color="Maroon" title="c9">f</font>, <font color="Maroon" title="c14">h'</font> = <font color="Maroon" title="c11">h</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E2:11_3_3"/><i><font color="Green" title="E17">A16</font></i>: 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font>
 <b>and </b><br/><a NAME="E3:11_3_3"/><i><font color="Green" title="E18">A17</font></i>: 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f</font>
 ;<br/>

<i><font color="Green" title="E19">A18</font></i>: the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">h'</font> = 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E5:11"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="nattra_1.html#E2:11_3_3"><i><font color="Green" title="E17">A16</font></i></a></i>
<br/>.= 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">g'</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E5:11"><i><font color="Green" title="E14">A5</font></i></a></i>
;<br/>
<i><font color="Green" title="E20">A19</font></i>: the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">g'</font> = 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E5:11"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="nattra_1.html#E3:11_3_3"><i><font color="Green" title="E18">A17</font></i></a></i>
<br/>.= 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">f'</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E5:11"><i><font color="Green" title="E14">A5</font></i></a></i>
;<br/>
<a NAME="E6:11_3_3"/><i><font color="Green" title="E21">A20</font></i>: 
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">h</font>,<font color="Maroon" title="c10">g</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c14">h'</font>,<font color="Maroon" title="c12">g'</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E2:11"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="nattra_1.html#E7:11"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="nattra_1.html#E2:11_3_3"><i><font color="Green" title="E17">A16</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<a NAME="E7:11_3_3"/><i><font color="Green" title="E22">A21</font></i>: 
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c12">g'</font>,<font color="Maroon" title="c13">f'</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E2:11"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="nattra_1.html#E7:11"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="nattra_1.html#E3:11_3_3"><i><font color="Green" title="E18">A17</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">gf</font> = the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c16">hg</font> = the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">h</font>,<font color="Maroon" title="c10">g</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <b>by </b><i><a class="txt" href="nattra_1.html#E7:11"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="nattra_1.html#E2:11_3_3"><i><font color="Green" title="E17">A16</font></i></a>, <a class="txt" href="nattra_1.html#E3:11_3_3"><i><font color="Green" title="E18">A17</font></i></a>, <a class="ref" href="partfun1.html#T27" title="PARTFUN1:th.27">PARTFUN1:27</a></i>;<br/>
<a NAME="E9:11_3_3"/>
( <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c12">g'</font>,<font color="Maroon" title="c13">f'</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">A</font> &amp; <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c14">h'</font>,<font color="Maroon" title="c12">g'</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><a class="txt" href="nattra_1.html#E4:11_3_3"><i><font color="Green" title="E19">A18</font></i></a>, <a class="txt" href="nattra_1.html#E5:11_3_3"><i><font color="Green" title="E20">A19</font></i></a>, <a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c17">gf'</font> = the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">g'</font>,<font color="Maroon" title="c13">f'</font>, <font color="Maroon" title="c18">hg'</font> = the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">h'</font>,<font color="Maroon" title="c12">g'</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="ref" href="partfun1.html#T27" title="PARTFUN1:th.27">PARTFUN1:27</a></i>;<br/>
<i><font color="Green" title="E23">A22</font></i>: the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">h</font> = 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">g'</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E5:11"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="nattra_1.html#E2:11_3_3"><i><font color="Green" title="E17">A16</font></i></a></i>
<br/>.= 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">gf'</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E5:11_3_3"><i><font color="Green" title="E20">A19</font></i></a>, <a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a></i>
<br/>.= 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">gf</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E5:11"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="nattra_1.html#E7:11_3_3"><i><font color="Green" title="E22">A21</font></i></a></i>
;<br/>
<i><font color="Green" title="E24">A23</font></i>: the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">hg</font> = 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">hg'</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E5:11"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="nattra_1.html#E6:11_3_3"><i><font color="Green" title="E21">A20</font></i></a></i>
<br/>.= 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">g'</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E4:11_3_3"><i><font color="Green" title="E19">A18</font></i></a>, <a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a></i>
<br/>.= 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E5:11"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="nattra_1.html#E3:11_3_3"><i><font color="Green" title="E18">A17</font></i></a></i>
;<br/>
<b>thus </b>the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">h</font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font></span>)</span> = 
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">h'</font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">g'</font>,<font color="Maroon" title="c13">f'</font></span>)</span>
<b>by </b><i><a class="txt" href="nattra_1.html#E2:11"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="nattra_1.html#E7:11"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="nattra_1.html#E7:11_3_3"><i><font color="Green" title="E22">A21</font></i></a>, <a class="txt" href="nattra_1.html#E11:11_3_3"><i><font color="Green" title="E23">A22</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">h'</font>,<font color="Maroon" title="c12">g'</font></span>)</span>,<font color="Maroon" title="c13">f'</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E4:11_3_3"><i><font color="Green" title="E19">A18</font></i></a>, <a class="txt" href="nattra_1.html#E5:11_3_3"><i><font color="Green" title="E20">A19</font></i></a>, <a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a></i>
<br/>.= 
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">h</font>,<font color="Maroon" title="c10">g</font></span>)</span>,<font color="Maroon" title="c9">f</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E2:11"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="nattra_1.html#E7:11"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="nattra_1.html#E6:11_3_3"><i><font color="Green" title="E21">A20</font></i></a>, <a class="txt" href="nattra_1.html#E12:11_3_3"><i><font color="Green" title="E24">A23</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c9">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #);<br/>

<b>reconsider </b><font color="Maroon" title="c10">b'</font> = <font color="Maroon" title="c9">b</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E5:11_3"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #)
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:11_3"/><b>then </b><i><font color="Green" title="E17">A24</font></i>: 
the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b'</font>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E3:11"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<b>hence </b>the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font></span>)</span> = 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b'</font></span>)</span>
<b>by </b><i><a class="txt" href="nattra_1.html#E5:11"><i><font color="Green" title="E14">A5</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c9">b</font>
<b>by </b><i><a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a></i>
;<br/><br/>

<b>thus </b>the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font></span>)</span> = 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b'</font></span>)</span>
<b>by </b><i><a class="txt" href="nattra_1.html#E5:11"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="nattra_1.html#E6:11_3"><i><font color="Green" title="E17">A24</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c9">b</font>
<b>by </b><i><a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a></i>
;<br/>

<b>thus </b><a NAME="E9:11_3"/>
for <font color="Olive" title="b1">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #)  st the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> holds <br/>the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font></span>)</span>,<font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">f</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #);<br/>

<b>assume </b><a NAME="E1:11_3_6"/><i><font color="Green" title="E18">A25</font></i>: 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c12">f'</font> = <font color="Maroon" title="c11">f</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E3:11_3_6"/><i><font color="Green" title="E19">A26</font></i>: 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b'</font>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E5:11"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="nattra_1.html#E1:11_3_6"><i><font color="Green" title="E18">A25</font></i></a></i>;<br/>
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">f</font> = 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">f'</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E5:11"><i><font color="Green" title="E14">A5</font></i></a></i>
<br/>.= 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b'</font></span>)</span>
<b>by </b><i><a class="txt" href="nattra_1.html#E3:11_3_6"><i><font color="Green" title="E19">A26</font></i></a>, <a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a></i>
<br/>.= 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font></span>)</span>
<b>by </b><i><a class="txt" href="nattra_1.html#E5:11"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="nattra_1.html#E6:11_3"><i><font color="Green" title="E17">A24</font></i></a></i>
;<br/>
<b>hence </b>the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font></span>)</span>,<font color="Maroon" title="c11">f</font> = 
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b'</font></span>)</span>,<font color="Maroon" title="c12">f'</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E2:11"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="nattra_1.html#E7:11"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="nattra_1.html#E6:11_3"><i><font color="Green" title="E17">A24</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon" title="c11">f</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E3:11_3_6"><i><font color="Green" title="E19">A26</font></i></a>, <a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a></i>
;<br/><br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c11">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #);<br/>

<b>assume </b><a NAME="E10:11_3"/><i><font color="Green" title="E18">A27</font></i>: 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c12">g'</font> = <font color="Maroon" title="c11">g</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E12:11_3"/><i><font color="Green" title="E19">A28</font></i>: 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b'</font>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E5:11"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="nattra_1.html#E10:11_3"><i><font color="Green" title="E18">A27</font></i></a></i>;<br/>
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font> = 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c1">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">g'</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E5:11"><i><font color="Green" title="E14">A5</font></i></a></i>
<br/>.= 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c1">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b'</font></span>)</span>
<b>by </b><i><a class="txt" href="nattra_1.html#E12:11_3"><i><font color="Green" title="E19">A28</font></i></a>, <a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a></i>
<br/>.= 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font></span>)</span>
<b>by </b><i><a class="txt" href="nattra_1.html#E5:11"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="nattra_1.html#E6:11_3"><i><font color="Green" title="E17">A24</font></i></a></i>
;<br/>
<b>hence </b>the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">g</font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">b</font></span>)</span> = 
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">g'</font>,<span class="p1">(<span class="default">the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c1">A</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">b'</font></span>)</span>
<b>by </b><i><a class="txt" href="nattra_1.html#E2:11"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="nattra_1.html#E7:11"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="nattra_1.html#E6:11_3"><i><font color="Green" title="E17">A24</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon" title="c11">g</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E12:11_3"><i><font color="Green" title="E19">A28</font></i></a>, <a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c9">C</font> =  <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) as   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a> ;<br/>
<a NAME="E10:11"/>
<font color="Maroon" title="c9">C</font> is    <a href="cat_2.html#M3" title="CAT_2:mode.3">Subcategory</a> of <font color="Maroon" title="c1">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:11_4"/>
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c9">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">A</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_2.html#D4" title="CAT_2:def.4">CAT_2:def 4</a><br/>

<b>thus </b><a NAME="E2:11_4"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c9">C</font><br/> for <font color="Olive" title="b3">a'</font>, <font color="Olive" title="b4">b'</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a'</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">b'</font> holds <br/> <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Olive" title="b3">a'</font>,<font color="Olive" title="b4">b'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">a</font>, <font color="Maroon" title="c11">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c9">C</font>;<br/><b>let </b><font color="Maroon" title="c12">a'</font>, <font color="Maroon" title="c13">b'</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font>;<br/>



<b>assume </b><a NAME="E1:11_4_1"/><i><font color="Green" title="E17">A29</font></i>: 
( <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a'</font> &amp; <font color="Maroon" title="c11">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b'</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:11_4_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c11">b</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c15">f</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c9">C</font><b> such that </b><br/><a NAME="E4:11_4_1"/><i><font color="Green" title="E18">A30</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">f</font>
 <b>and </b><br/><a NAME="E5:11_4_1"/><i><font color="Green" title="E19">A31</font></i>: 
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c16">f'</font> = <font color="Maroon" title="c15">f</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E7:11_4_1"/><i><font color="Green" title="E20">A32</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c16">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a'</font>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E5:11"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="nattra_1.html#E1:11_4_1"><i><font color="Green" title="E17">A29</font></i></a>, <a class="txt" href="nattra_1.html#E5:11_4_1"><i><font color="Green" title="E19">A31</font></i></a></i>;<br/>
<a NAME="E8:11_4_1"/>
 <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c16">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b'</font>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E5:11"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="nattra_1.html#E1:11_4_1"><i><font color="Green" title="E17">A29</font></i></a>, <a class="txt" href="nattra_1.html#E5:11_4_1"><i><font color="Green" title="E19">A31</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:11_4_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c12">a'</font>,<font color="Maroon" title="c13">b'</font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E4:11_4_1"><i><font color="Green" title="E18">A30</font></i></a>, <a class="txt" href="nattra_1.html#E7:11_4_1"><i><font color="Green" title="E20">A32</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:11_4"/>
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c9">C</font> <a href="funct_4.html#NR1" title="FUNCT_4:NR.1">&lt;=</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E2:11"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="relat_1.html#T88" title="RELAT_1:th.88">RELAT_1:88</a></i>;<br/>

<b>let </b><font color="Maroon" title="c10">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c9">C</font>;<br/><b>let </b><font color="Maroon" title="c11">a'</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">A</font>;<br/>



<b>assume </b><a NAME="E4:11_4"/><i><font color="Green" title="E17">A33</font></i>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">a'</font>
 ;<br/>

<a NAME="E5:11_4"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c9">C</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:11_4"/>
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c11">a'</font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E3:11"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="nattra_1.html#E4:11_4"><i><font color="Green" title="E17">A33</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:11"/>
 <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">DOM</font>,<font color="Maroon" title="c5">COD</font>,<font color="Maroon" title="c6">COMP</font>,<font color="Maroon" title="c7">ID</font> #) is    <a href="cat_2.html#M3" title="CAT_2:mode.3">Subcategory</a> of <font color="Maroon" title="c1">A</font>
 ;<br/>


</div>
