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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">o</font>, <font color="Olive" title="b2">m</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds   <a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">m</font></span>)</span>,<font color="Maroon" title="c1">A</font> <a href="isocat_1.html#NR3" title="ISOCAT_1:NR.3">~=</a> <font color="Maroon" title="c1">A</font></span><br/><b>let </b><font color="Maroon" title="c2">o</font>, <font color="Maroon" title="c3">m</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c1">A</font> <a href="isocat_1.html#NR3" title="ISOCAT_1:NR.3">~=</a> <font color="Maroon" title="c1">A</font></span><br/>





<b>reconsider </b><font color="Maroon" title="c4">a</font> = <font color="Maroon" title="c2">o</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span> <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>take </b><font color="Maroon" title="c5">F</font> = <font color="Maroon" title="c4">a</font> <a href="isocat_2.html#K2" title="ISOCAT_2:func.2">|-&gt;</a> <font color="Maroon" title="c1">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="isocat_1.html#D4" title="ISOCAT_1:def.4">ISOCAT_1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">F</font> is <a href="cat_1.html#V9" title="CAT_1:attr.9">isomorphic</a></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <span class="p2">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c1">A</font></span>)</span> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <span class="p2">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c1">A</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font> implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> )</span><br/><a NAME="E1:12_1"/><i><font color="Green" title="E11">A1</font></i>: 
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <span class="p2">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></i>;<br/><b>assume </b><a NAME="E2:12_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <span class="p2">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c1">A</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <span class="p2">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c1">A</font></span>)</span> &amp; <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font> implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> )</span><br/><b>then consider </b><font color="Maroon" title="c8">F1</font>, <font color="Maroon" title="c9">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of  <a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c10">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c8">F1</font>,<font color="Maroon" title="c9">F2</font><b> such that </b><br/><a NAME="E4:12_1"/><i><font color="Green" title="E12">A2</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">F1</font>,<font color="Maroon" title="c9">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c10">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c8">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c9">F2</font> )
 <b>by </b><i><a class="txt" href="isocat_2.html#E1:12_1"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="nattra_1.html#D16" title="NATTRA_1:def.16">NATTRA_1:def 16</a></i>;<br/><b>assume </b><a NAME="E5:12_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <span class="p2">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c1">A</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font> implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> )</span><br/><b>then consider </b><font color="Maroon" title="c11">F1'</font>, <font color="Maroon" title="c12">F2'</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of  <a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c13">t'</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c11">F1'</font>,<font color="Maroon" title="c12">F2'</font><b> such that </b><br/><a NAME="E7:12_1"/><i><font color="Green" title="E13">A3</font></i>: 
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">F1'</font>,<font color="Maroon" title="c12">F2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c13">t'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c11">F1'</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c12">F2'</font> )
 <b>by </b><i><a class="txt" href="isocat_2.html#E1:12_1"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="nattra_1.html#D16" title="NATTRA_1:def.16">NATTRA_1:def 16</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">s</font> = <font color="Maroon" title="c10">t</font>, <font color="Maroon" title="c15">s'</font> = <font color="Maroon" title="c13">t'</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">A</font> ;<br/><b>reconsider </b><font color="Maroon" title="c16">G1</font> = <font color="Maroon" title="c8">F1</font>, <font color="Maroon" title="c17">G1'</font> = <font color="Maroon" title="c11">F1'</font>, <font color="Maroon" title="c18">G2</font> = <font color="Maroon" title="c9">F2</font>, <font color="Maroon" title="c19">G2'</font> = <font color="Maroon" title="c12">F2'</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">A</font> ;<br/><a NAME="E10:12_1"/><i><font color="Green" title="E14">A4</font></i>: 
( <font color="Maroon" title="c8">F1</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <font color="Maroon" title="c9">F2</font> &amp; <font color="Maroon" title="c11">F1'</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <font color="Maroon" title="c12">F2'</font> )
 <b>by </b><i><a class="txt" href="isocat_2.html#E4:12_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="isocat_2.html#E7:12_1"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="nattra_1.html#D7" title="NATTRA_1:def.7">NATTRA_1:def 7</a></i>;<br/><b>assume </b><a NAME="E11:12_1"/>
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font></span><br/><b>then </b><i><font color="Green" title="E15">A5</font></i>: <font color="Maroon" title="c10">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">a</font> = 
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font>
<b>by </b><i><a class="txt" href="isocat_2.html#E4:12_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="isocat_2.html#D1" target="_self" title="ISOCAT_2:def.1">Def1</a></i>
<br/>.= 
<font color="Maroon" title="c13">t'</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="txt" href="isocat_2.html#E7:12_1"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="isocat_2.html#D1" target="_self" title="ISOCAT_2:def.1">Def1</a></i>
;<br/><b>then </b><i><font color="Green" title="E16">A6</font></i>: <font color="Maroon" title="c14">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font> = 
<font color="Maroon" title="c13">t'</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="txt" href="isocat_2.html#E10:12_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="nattra_1.html#D5" title="NATTRA_1:def.5">NATTRA_1:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c15">s'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="txt" href="isocat_2.html#E10:12_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="nattra_1.html#D5" title="NATTRA_1:def.5">NATTRA_1:def 5</a></i>
;<br/><a NAME="E14:12_1"/><i><font color="Green" title="E17">A7</font></i>: 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E15:12_1"/>
(  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c4">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">F2</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F1'</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c4">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">F2'</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="txt" href="isocat_2.html#E10:12_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="nattra_1.html#D2" title="NATTRA_1:def.2">NATTRA_1:def 2</a></i>;<br/><a NAME="E16:12_1"/><b>then </b><i><font color="Green" title="E18">A8</font></i>: 
( <font color="Maroon" title="c8">F1</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">F1'</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c9">F2</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F2'</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c4">a</font> )
 <b>by </b><i><a class="txt" href="isocat_2.html#E12:12_1"><i><font color="Green" title="E15">A5</font></i></a>, <a class="ref" href="cat_1.html#T24" title="CAT_1:th.24">CAT_1:24</a></i>;<br/><b>then </b><i><font color="Green" title="E19">A9</font></i>: <font color="Maroon" title="c16">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">a</font></span>)</span> = 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F1'</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c4">a</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T108" title="CAT_1:th.108">CAT_1:108</a></i>
<br/>.= 
<font color="Maroon" title="c17">G1'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">a</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T108" title="CAT_1:th.108">CAT_1:108</a></i>
;<br/><font color="Maroon" title="c18">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">a</font></span>)</span> = 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F2'</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c4">a</font></span>)</span>
<b>by </b><i><a class="txt" href="isocat_2.html#E16:12_1"><i><font color="Green" title="E18">A8</font></i></a>, <a class="ref" href="cat_1.html#T108" title="CAT_1:th.108">CAT_1:108</a></i>
<br/>.= 
<font color="Maroon" title="c19">G2'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">a</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T108" title="CAT_1:th.108">CAT_1:108</a></i>
;<br/><a NAME="E19:12_1"/><b>then </b>
( <font color="Maroon" title="c8">F1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c11">F1'</font> &amp; <font color="Maroon" title="c9">F2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c12">F2'</font> )
 <b>by </b><i><a class="txt" href="isocat_2.html#E14:12_1"><i><font color="Green" title="E17">A7</font></i></a>, <a class="txt" href="isocat_2.html#E17:12_1"><i><font color="Green" title="E19">A9</font></i></a>, <a class="ref" href="isocat_2.html#T3" target="_self" title="ISOCAT_2:th.3">Th3</a></i>;<br/><b>hence </b><a NAME="E20:12_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font>
 <b>by </b><i><a class="txt" href="isocat_2.html#E4:12_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="isocat_2.html#E7:12_1"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="isocat_2.html#E13:12_1"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="isocat_2.html#T3" target="_self" title="ISOCAT_2:th.3">Th3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:12"/>
<font color="Maroon" title="c5">F</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_2.html#T25" title="FUNCT_2:th.25">FUNCT_2:25</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="isocat_1.html#D3" title="ISOCAT_1:def.3">ISOCAT_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">F</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 <font color="Maroon" title="c1">A</font></span><br/>

<b>thus </b><a NAME="E4:12"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">A</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">F</font></span><br/>

<b>let </b><font color="Maroon" title="c6">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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">A</font> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">F</font> )</span><br/>

<b>assume </b><a NAME="E5:12"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">F</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c7">f</font> = <font color="Maroon" title="c6">x</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">A</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">F1</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span>, <font color="Maroon" title="c9">F2</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">A</font> ;<br/>
<a NAME="E8:12"/><i><font color="Green" title="E11">A13</font></i>: 
for <font color="Olive" title="b1">c</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>  ex <font color="Olive" title="b2">d</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="Maroon" title="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Olive" title="b1">c</font></span>)</span> <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="Olive" title="b2">d</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<div><i><font color="Green" title="E12">A14</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>now </b></a><div class="add"><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 <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span> )</span><br/><b>thus </b><font color="Maroon" title="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span> = 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c7">f</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span></span><br/><b>thus </b><font color="Maroon" title="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span> = 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c7">f</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E13">A15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">h</font>, <font color="Maroon" title="c11">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c10">h</font> implies <font color="Maroon" title="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c10">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">h</font></span>)</span> )</span><br/><b>assume </b><a NAME="E1:12_3"/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c10">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c10">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">h</font></span>)</span></span><br/><a NAME="E2:12_3"/><i><font color="Green" title="E14">A16</font></i>: 
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c7">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="cat_1.html#T56" title="CAT_1:th.56">CAT_1:56</a></i>;<br/><b>thus </b><font color="Maroon" title="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c10">h</font></span>)</span> = 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c7">f</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T59" title="CAT_1:th.59">CAT_1:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="isocat_2.html#E2:12_3"><i><font color="Green" title="E14">A16</font></i></a>, <a class="ref" href="cat_1.html#D13" title="CAT_1:def.13">CAT_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">h</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">h</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E11:12"/><i><font color="Green" title="E14">A17</font></i>: 
for <font color="Olive" title="b1">c</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>  ex <font color="Olive" title="b2">d</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="Maroon" title="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Olive" title="b1">c</font></span>)</span> <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="Olive" title="b2">d</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<div><i><font color="Green" title="E15">A18</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_4"><b>now </b></a><div class="add"><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 <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span> )</span><br/><b>thus </b><font color="Maroon" title="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span> = 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">f</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span></span><br/><b>thus </b><font color="Maroon" title="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span> = 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">f</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T44" title="CAT_1:th.44">CAT_1:44</a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">h</font>, <font color="Maroon" title="c11">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c10">h</font> implies <font color="Maroon" title="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c10">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">h</font></span>)</span> )</span><br/><b>assume </b><a NAME="E1:12_5"/>
 <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c10">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c10">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">h</font></span>)</span></span><br/><a NAME="E2:12_5"/><i><font color="Green" title="E16">A19</font></i>: 
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="cat_1.html#T56" title="CAT_1:th.56">CAT_1:56</a></i>;<br/><b>thus </b><font color="Maroon" title="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c10">h</font></span>)</span> = 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">f</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T59" title="CAT_1:th.59">CAT_1:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="isocat_2.html#E2:12_5"><i><font color="Green" title="E16">A19</font></i></a>, <a class="ref" href="cat_1.html#D13" title="CAT_1:def.13">CAT_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">h</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">h</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c10">F1</font> = <font color="Maroon" title="c8">F1</font>, <font color="Maroon" title="c11">F2</font> = <font color="Maroon" title="c9">F2</font> as    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of  <a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="isocat_2.html#E8:12"><i><font color="Green" title="E11">A13</font></i></a>, <a class="txt" href="isocat_2.html#E9:12"><i><font color="Green" title="E12">A14</font></i></a>, <a class="txt" href="isocat_2.html#E10:12"><i><font color="Green" title="E13">A15</font></i></a>, <a class="txt" href="isocat_2.html#E11:12"><i><font color="Green" title="E14">A17</font></i></a>, <a class="txt" href="isocat_2.html#E12:12"><i><font color="Green" title="E15">A18</font></i></a>, <a class="ref" href="cat_1.html#T96" title="CAT_1:th.96">CAT_1:96</a></i>;<br/>
<a NAME="E15:12"/><i><font color="Green" title="E16">A20</font></i>: 
for <font color="Olive" title="b1">b</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span> holds <br/> ( <font color="Maroon" title="c10">F1</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c7">f</font> &amp; <font color="Maroon" title="c11">F2</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">f</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">F1</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c7">f</font> &amp; <font color="Maroon" title="c11">F2</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">f</font> )</span><br/>

<a NAME="E1:12_6"/>
( <font color="Maroon" title="c10">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c7">f</font></span>)</span> &amp; <font color="Maroon" title="c11">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">f</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E2:12_6"/><b>then </b>
(  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c7">f</font></span>)</span> &amp;  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F2</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">f</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T108" title="CAT_1:th.108">CAT_1:108</a></i>;<br/>
<b>hence </b><a NAME="E3:12_6"/>
( <font color="Maroon" title="c10">F1</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c7">f</font> &amp; <font color="Maroon" title="c11">F2</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">f</font> )
 <b>by </b><i><a class="ref" href="cat_1.html#T45" title="CAT_1:th.45">CAT_1:45</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E17">A21</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c12">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">F2</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><a NAME="E1:12_7"/>
( <font color="Maroon" title="c10">F1</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c7">f</font> &amp; <font color="Maroon" title="c11">F2</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">f</font> )
 <b>by </b><i><a class="txt" href="isocat_2.html#E15:12"><i><font color="Green" title="E16">A20</font></i></a></i>;<br/><b>hence </b><a NAME="E2:12_7"/>
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c12">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">F2</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c12">b</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="cat_1.html#T19" title="CAT_1:th.19">CAT_1:19</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E17:12"/><b>then </b><i><font color="Green" title="E18">A22</font></i>: 
<font color="Maroon" title="c10">F1</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <font color="Maroon" title="c11">F2</font>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D2" title="NATTRA_1:def.2">NATTRA_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">t</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c7">f</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 <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c1">A</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">b</font> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c10">F1</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c11">F2</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c13">b</font></span><br/><a NAME="E1:12_8"/><i><font color="Green" title="E19">A24</font></i>: 
<font color="Maroon" title="c12">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><a NAME="E2:12_8"/>
( <font color="Maroon" title="c10">F1</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c13">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c7">f</font> &amp; <font color="Maroon" title="c11">F2</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c13">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">f</font> )
 <b>by </b><i><a class="txt" href="isocat_2.html#E15:12"><i><font color="Green" title="E16">A20</font></i></a></i>;<br/><a NAME="E3:12_8"/><b>then </b>
<font color="Maroon" title="c12">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">b</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> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c13">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">F2</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c13">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="isocat_2.html#E1:12_8"><i><font color="Green" title="E19">A24</font></i></a></i>;<br/><b>hence </b><a NAME="E4:12_8"/>
<font color="Maroon" title="c12">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">b</font> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c10">F1</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c11">F2</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c13">b</font>
 <b>by </b><i><a class="ref" href="cat_1.html#D7" title="CAT_1:def.7">CAT_1:def 7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c13">t</font> = <font color="Maroon" title="c12">t</font> as    <a href="nattra_1.html#M1" title="NATTRA_1:mode.1">transformation</a> of <font color="Maroon" title="c10">F1</font>,<font color="Maroon" title="c11">F2</font> <b>by </b><i><a class="txt" href="isocat_2.html#E17:12"><i><font color="Green" title="E18">A22</font></i></a>, <a class="ref" href="nattra_1.html#D3" title="NATTRA_1:def.3">NATTRA_1:def 3</a></i>;<br/>
<a NAME="E21:12"/><i><font color="Green" title="E19">A25</font></i>: 
for <font color="Olive" title="b1">b</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span> holds  <font color="Maroon" title="c13">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font></span><br/>

<b>thus </b><font color="Maroon" title="c7">f</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">b</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<font color="Maroon" title="c13">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">b</font>
<b>by </b><i><a class="txt" href="isocat_2.html#E17:12"><i><font color="Green" title="E18">A22</font></i></a>, <a class="ref" href="nattra_1.html#D5" title="NATTRA_1:def.5">NATTRA_1:def 5</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E20">A26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_10"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">b1</font>, <font color="Maroon" title="c15">b2</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c14">b1</font>,<font color="Maroon" title="c15">b2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies for <font color="Olive" title="b1">g</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c14">b1</font>,<font color="Maroon" title="c15">b2</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c13">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c15">b2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">b1</font></span>)</span> )</span><br/><b>assume </b><a NAME="E1:12_10"/><i><font color="Green" title="E21">A27</font></i>: 
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c14">b1</font>,<font color="Maroon" title="c15">b2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">g</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c14">b1</font>,<font color="Maroon" title="c15">b2</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c13">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c15">b2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">b1</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c16">g</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c14">b1</font>,<font color="Maroon" title="c15">b2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c13">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c15">b2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">b1</font></span>)</span></span><br/><a NAME="E2:12_10"/><i><font color="Green" title="E22">A28</font></i>: 
<font color="Maroon" title="c13">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font>
 <b>by </b><i><a class="txt" href="isocat_2.html#E21:12"><i><font color="Green" title="E19">A25</font></i></a></i>;<br/><a NAME="E3:12_10"/><i><font color="Green" title="E23">A29</font></i>: 
<font color="Maroon" title="c13">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c15">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font>
 <b>by </b><i><a class="txt" href="isocat_2.html#E21:12"><i><font color="Green" title="E19">A25</font></i></a></i>;<br/><a NAME="E4:12_10"/><i><font color="Green" title="E24">A30</font></i>: 
<font color="Maroon" title="c16">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E5:12_10"/><i><font color="Green" title="E25">A31</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E6:12_10"/><i><font color="Green" title="E26">A32</font></i>: 
(  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c14">b1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">F2</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c14">b1</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F2</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c14">b1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">F2</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c15">b2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="txt" href="isocat_2.html#E16:12"><i><font color="Green" title="E17">A21</font></i></a>, <a class="txt" href="isocat_2.html#E1:12_10"><i><font color="Green" title="E21">A27</font></i></a>, <a class="ref" href="cat_1.html#T124" title="CAT_1:th.124">CAT_1:124</a></i>;<br/><a NAME="E7:12_10"/><i><font color="Green" title="E27">A33</font></i>: 
(  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c15">b2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">F2</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c15">b2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c14">b1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c15">b2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="txt" href="isocat_2.html#E16:12"><i><font color="Green" title="E17">A21</font></i></a>, <a class="txt" href="isocat_2.html#E1:12_10"><i><font color="Green" title="E21">A27</font></i></a>, <a class="ref" href="cat_1.html#T124" title="CAT_1:th.124">CAT_1:124</a></i>;<br/><i><font color="Green" title="E28">A34</font></i>: <font color="Maroon" title="c11">F2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c16">g</font> = 
<font color="Maroon" title="c11">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font>
<b>by </b><i><a class="txt" href="isocat_2.html#E1:12_10"><i><font color="Green" title="E21">A27</font></i></a>, <a class="txt" href="isocat_2.html#E4:12_10"><i><font color="Green" title="E24">A30</font></i></a>, <a class="ref" href="nattra_1.html#D1" title="NATTRA_1:def.1">NATTRA_1:def 1</a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">f</font></span>)</span>
<b>by </b><i><a class="txt" href="isocat_2.html#E5:12_10"><i><font color="Green" title="E25">A31</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><i><font color="Green" title="E29">A35</font></i>: <font color="Maroon" title="c10">F1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c16">g</font> = 
<font color="Maroon" title="c10">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">m</font>
<b>by </b><i><a class="txt" href="isocat_2.html#E1:12_10"><i><font color="Green" title="E21">A27</font></i></a>, <a class="txt" href="isocat_2.html#E4:12_10"><i><font color="Green" title="E24">A30</font></i></a>, <a class="ref" href="nattra_1.html#D1" title="NATTRA_1:def.1">NATTRA_1:def 1</a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c7">f</font></span>)</span>
<b>by </b><i><a class="txt" href="isocat_2.html#E5:12_10"><i><font color="Green" title="E25">A31</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c13">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c15">b2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c16">g</font></span>)</span> = 
<font color="Maroon" title="c7">f</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c16">g</font></span>)</span>
<b>by </b><i><a class="txt" href="isocat_2.html#E3:12_10"><i><font color="Green" title="E23">A29</font></i></a>, <a class="txt" href="isocat_2.html#E7:12_10"><i><font color="Green" title="E27">A33</font></i></a>, <a class="ref" href="cat_1.html#D13" title="CAT_1:def.13">CAT_1:def 13</a></i>
<br/>.= 
<font color="Maroon" title="c7">f</font>
<b>by </b><i><a class="txt" href="isocat_2.html#E9:12_10"><i><font color="Green" title="E29">A35</font></i></a>, <a class="ref" href="cat_1.html#T47" title="CAT_1:th.47">CAT_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">F2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c7">f</font>
<b>by </b><i><a class="txt" href="isocat_2.html#E8:12_10"><i><font color="Green" title="E28">A34</font></i></a>, <a class="ref" href="cat_1.html#T46" title="CAT_1:th.46">CAT_1:46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">F2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Maroon" title="c16">g</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c14">b1</font></span>)</span>
<b>by </b><i><a class="txt" href="isocat_2.html#E2:12_10"><i><font color="Green" title="E22">A28</font></i></a>, <a class="txt" href="isocat_2.html#E6:12_10"><i><font color="Green" title="E26">A32</font></i></a>, <a class="ref" href="cat_1.html#D13" title="CAT_1:def.13">CAT_1:def 13</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E23:12"/><i><font color="Green" title="E21">A36</font></i>: 
<font color="Maroon" title="c10">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c11">F2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_11"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:12_11"/>
<font color="Maroon" title="c10">F1</font> <a href="nattra_1.html#R1" title="NATTRA_1:pred.1">is_transformable_to</a> <font color="Maroon" title="c11">F2</font>
 <b>by </b><i><a class="txt" href="isocat_2.html#E16:12"><i><font color="Green" title="E17">A21</font></i></a>, <a class="ref" href="nattra_1.html#D2" title="NATTRA_1:def.2">NATTRA_1:def 2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D7" title="NATTRA_1:def.7">NATTRA_1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="nattra_1.html#M1" title="NATTRA_1:mode.1">transformation</a> of <font color="Maroon" title="c10">F1</font>,<font color="Maroon" title="c11">F2</font> st <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>  holds <br/>(  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or for <font color="Olive">b<sub>4</sub></font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> holds  <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> )</span><br/>

<b>thus </b><a NAME="E2:12_11"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="nattra_1.html#M1" title="NATTRA_1:mode.1">transformation</a> of <font color="Maroon" title="c10">F1</font>,<font color="Maroon" title="c11">F2</font> st <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>  holds <br/>(  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or for <font color="Olive">b<sub>4</sub></font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> holds  <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">F1</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F2</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="isocat_2.html#E22:12"><i><font color="Green" title="E20">A26</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c14">t</font> = <font color="Maroon" title="c13">t</font> as    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c10">F1</font>,<font color="Maroon" title="c11">F2</font> <b>by </b><i><a class="txt" href="isocat_2.html#E22:12"><i><font color="Green" title="E20">A26</font></i></a>, <a class="ref" href="nattra_1.html#D8" title="NATTRA_1:def.8">NATTRA_1:def 8</a></i>;<br/>
<a NAME="E25:12"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">F1</font>,<font color="Maroon" title="c11">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c14">t</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="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <span class="p1">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="isocat_2.html#E23:12"><i><font color="Green" title="E21">A36</font></i></a>, <a class="ref" href="nattra_1.html#D16" title="NATTRA_1:def.16">NATTRA_1:def 16</a></i>;<br/>
<a NAME="E26:12"/><b>then </b><i><font color="Green" title="E22">A37</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">F1</font>,<font color="Maroon" title="c11">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c14">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" title="NATTRA_1:func.12">Functors</a> <span class="p2">(<span class="default"><a href="cat_1.html#K8" title="CAT_1:func.8">1Cat</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">m</font></span>)</span>,<font color="Maroon" title="c1">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D18" title="NATTRA_1:def.18">NATTRA_1:def 18</a></i>;<br/>
<font color="Maroon" title="c5">F</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">F1</font>,<font color="Maroon" title="c11">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c14">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<font color="Maroon" title="c14">t</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="txt" href="isocat_2.html#E23:12"><i><font color="Green" title="E21">A36</font></i></a>, <a class="ref" href="isocat_2.html#D1" target="_self" title="ISOCAT_2:def.1">Def1</a></i>
<br/>.= 
<font color="Maroon" title="c7">f</font>
<b>by </b><i><a class="txt" href="isocat_2.html#E21:12"><i><font color="Green" title="E19">A25</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E28:12"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">F</font>
 <b>by </b><i><a class="txt" href="isocat_2.html#E26:12"><i><font color="Green" title="E22">A37</font></i></a>, <a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
