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<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c3">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/><b>let </b><font color="Maroon" title="c4">F1</font>, <font color="Maroon" title="c5">G1</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>;<br/><b>let </b><font color="Maroon" title="c6">F2</font>, <font color="Maroon" title="c7">G2</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:58"/><i><font color="Green" title="E50">A1</font></i>: 
<font color="Maroon" title="c4">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c5">G1</font>
 <b>and </b><br/><a NAME="E2:58"/><i><font color="Green" title="E51">A2</font></i>: 
<font color="Maroon" title="c6">F2</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c7">G2</font>
 ;<br/>

<a NAME="E3:58"/><i><font color="Green" title="E52">A3</font></i>: 
( <font color="Maroon" title="c4">F1</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <font color="Maroon" title="c5">G1</font> &amp; <font color="Maroon" title="c6">F2</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <font color="Maroon" title="c7">G2</font> )
 
<b>by </b><i><a class="txt" href="isocat_2.html#E1:58"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="isocat_2.html#E2:58"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="nattra_1.html#D7" title="NATTRA_1:def.7">NATTRA_1:def 7</a></i>;<br/>
<b>let </b><font color="Maroon" title="c8">t1</font> be    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">G1</font>;<br/><b>let </b><font color="Maroon" title="c9">t2</font> be    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c6">F2</font>,<font color="Maroon" title="c7">G2</font>;<br/>



<b>reconsider </b><font color="Maroon" title="c10">s</font> = <font color="Maroon" title="c8">t1</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c1">A</font>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">B</font> ;<br/>
<a NAME="E5:58"/>
<span class="p1"><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">&lt;:</a><span class="default"><font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c6">F2</font></span><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">:&gt;</a></span> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <span class="p1"><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">&lt;:</a><span class="default"><font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c7">G2</font></span><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">:&gt;</a></span>
 
<b>by </b><i><a class="txt" href="isocat_2.html#E1:58"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="isocat_2.html#E2:58"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="isocat_2.html#T46" target="_self" title="ISOCAT_2:th.46">Th46</a></i>;<br/>
<a NAME="E6:58"/><b>then </b><i><font color="Green" title="E53">A4</font></i>: 
<span class="p1"><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">&lt;:</a><span class="default"><font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c6">F2</font></span><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">:&gt;</a></span> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <span class="p1"><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">&lt;:</a><span class="default"><font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c7">G2</font></span><a href="isocat_2.html#K10" title="ISOCAT_2:func.10">:&gt;</a></span>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D7" title="NATTRA_1:def.7">NATTRA_1:def 7</a></i>;<br/>
<b>thus </b> <a href="isocat_2.html#K13" title="ISOCAT_2:func.13">Pr1</a> <span class="p1"><a href="isocat_2.html#K16" title="ISOCAT_2:func.16">&lt;:</a><span class="default"><font color="Maroon" title="c8">t1</font>,<font color="Maroon" title="c9">t2</font></span><a href="isocat_2.html#K16" title="ISOCAT_2:func.16">:&gt;</a></span> = 
<span class="p1">(<span class="default"><a href="isocat_2.html#K8" title="ISOCAT_2:func.8">pr1</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="isocat_1.html#K2" title="ISOCAT_1:func.2">*</a> <span class="p1"><a href="isocat_2.html#K16" title="ISOCAT_2:func.16">&lt;:</a><span class="default"><font color="Maroon" title="c8">t1</font>,<font color="Maroon" title="c9">t2</font></span><a href="isocat_2.html#K16" title="ISOCAT_2:func.16">:&gt;</a></span>
<b>by </b><i><a class="txt" href="isocat_2.html#E1:58"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="isocat_2.html#E2:58"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="isocat_2.html#T46" target="_self" title="ISOCAT_2:th.46">Th46</a>, <a class="ref" href="isocat_1.html#D7" title="ISOCAT_1:def.7">ISOCAT_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="isocat_2.html#K8" title="ISOCAT_2:func.8">pr1</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1"><a href="isocat_2.html#K16" title="ISOCAT_2:func.16">&lt;:</a><span class="default"><font color="Maroon" title="c8">t1</font>,<font color="Maroon" title="c9">t2</font></span><a href="isocat_2.html#K16" title="ISOCAT_2:func.16">:&gt;</a></span>
<b>by </b><i><a class="txt" href="isocat_2.html#E6:58"><i><font color="Green" title="E53">A4</font></i></a>, <a class="ref" href="isocat_1.html#D5" title="ISOCAT_1:def.5">ISOCAT_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="isocat_2.html#K8" title="ISOCAT_2:func.8">pr1</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1"><a href="isocat_2.html#K15" title="ISOCAT_2:func.15">&lt;:</a><span class="default"><font color="Maroon" title="c8">t1</font>,<font color="Maroon" title="c9">t2</font></span><a href="isocat_2.html#K15" title="ISOCAT_2:func.15">:&gt;</a></span>
<b>by </b><i><a class="txt" href="isocat_2.html#E1:58"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="isocat_2.html#E2:58"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="isocat_2.html#D12" target="_self" title="ISOCAT_2:def.12">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="isocat_2.html#K8" title="ISOCAT_2:func.8">pr1</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1"><a href="funct_3.html#K14" title="FUNCT_3:func.14">&lt;:</a><span class="default"><font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c9">t2</font></span><a href="funct_3.html#K14" title="FUNCT_3:func.14">:&gt;</a></span>
<b>by </b><i><a class="txt" href="isocat_2.html#E3:58"><i><font color="Green" title="E52">A3</font></i></a>, <a class="ref" href="isocat_2.html#D11" target="_self" title="ISOCAT_2:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">B</font>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">C</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1"><a href="funct_3.html#K14" title="FUNCT_3:func.14">&lt;:</a><span class="default"><font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c9">t2</font></span><a href="funct_3.html#K14" title="FUNCT_3:func.14">:&gt;</a></span>
<b>by </b><i><a class="ref" href="cat_2.html#D10" title="CAT_2:def.10">CAT_2:def 10</a></i>
<br/>.= 
<font color="Maroon" title="c8">t1</font>
<b>by </b><i><a class="ref" href="funct_3.html#T82" title="FUNCT_3:th.82">FUNCT_3:82</a></i>
;<br/>

<b>thus </b> <a href="isocat_2.html#K14" title="ISOCAT_2:func.14">Pr2</a> <span class="p1"><a href="isocat_2.html#K16" title="ISOCAT_2:func.16">&lt;:</a><span class="default"><font color="Maroon" title="c8">t1</font>,<font color="Maroon" title="c9">t2</font></span><a href="isocat_2.html#K16" title="ISOCAT_2:func.16">:&gt;</a></span> = 
<span class="p1">(<span class="default"><a href="isocat_2.html#K9" title="ISOCAT_2:func.9">pr2</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="isocat_1.html#K2" title="ISOCAT_1:func.2">*</a> <span class="p1"><a href="isocat_2.html#K16" title="ISOCAT_2:func.16">&lt;:</a><span class="default"><font color="Maroon" title="c8">t1</font>,<font color="Maroon" title="c9">t2</font></span><a href="isocat_2.html#K16" title="ISOCAT_2:func.16">:&gt;</a></span>
<b>by </b><i><a class="txt" href="isocat_2.html#E1:58"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="isocat_2.html#E2:58"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="isocat_2.html#T46" target="_self" title="ISOCAT_2:th.46">Th46</a>, <a class="ref" href="isocat_1.html#D7" title="ISOCAT_1:def.7">ISOCAT_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="isocat_2.html#K9" title="ISOCAT_2:func.9">pr2</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1"><a href="isocat_2.html#K16" title="ISOCAT_2:func.16">&lt;:</a><span class="default"><font color="Maroon" title="c8">t1</font>,<font color="Maroon" title="c9">t2</font></span><a href="isocat_2.html#K16" title="ISOCAT_2:func.16">:&gt;</a></span>
<b>by </b><i><a class="txt" href="isocat_2.html#E6:58"><i><font color="Green" title="E53">A4</font></i></a>, <a class="ref" href="isocat_1.html#D5" title="ISOCAT_1:def.5">ISOCAT_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="isocat_2.html#K9" title="ISOCAT_2:func.9">pr2</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1"><a href="isocat_2.html#K15" title="ISOCAT_2:func.15">&lt;:</a><span class="default"><font color="Maroon" title="c8">t1</font>,<font color="Maroon" title="c9">t2</font></span><a href="isocat_2.html#K15" title="ISOCAT_2:func.15">:&gt;</a></span>
<b>by </b><i><a class="txt" href="isocat_2.html#E1:58"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="isocat_2.html#E2:58"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="isocat_2.html#D12" target="_self" title="ISOCAT_2:def.12">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="isocat_2.html#K9" title="ISOCAT_2:func.9">pr2</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1"><a href="funct_3.html#K14" title="FUNCT_3:func.14">&lt;:</a><span class="default"><font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c9">t2</font></span><a href="funct_3.html#K14" title="FUNCT_3:func.14">:&gt;</a></span>
<b>by </b><i><a class="txt" href="isocat_2.html#E3:58"><i><font color="Green" title="E52">A3</font></i></a>, <a class="ref" href="isocat_2.html#D11" target="_self" title="ISOCAT_2:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">B</font>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">C</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1"><a href="funct_3.html#K14" title="FUNCT_3:func.14">&lt;:</a><span class="default"><font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c9">t2</font></span><a href="funct_3.html#K14" title="FUNCT_3:func.14">:&gt;</a></span>
<b>by </b><i><a class="ref" href="cat_2.html#D11" title="CAT_2:def.11">CAT_2:def 11</a></i>
<br/>.= 
<font color="Maroon" title="c9">t2</font>
<b>by </b><i><a class="ref" href="funct_3.html#T82" title="FUNCT_3:th.82">FUNCT_3:82</a></i>
;<br/>


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