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<b>let </b><font color="Maroon" title="c3">C1</font>, <font color="Maroon" title="c4">C2</font> be  <a href="cat_1.html#V1" title="CAT_1:attr.1">strict</a> <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:58_1_2"/><i><font color="Green" title="E47">A64</font></i>: 
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c3">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>
 <b>and </b><br/><a NAME="E2:58_1_2"/><i><font color="Green" title="E48">A65</font></i>: 
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>
 <b>and </b><br/><a NAME="E3:58_1_2"/><i><font color="Green" title="E49">A66</font></i>: 
for <font color="Olive" title="b1">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C1</font> holds <br/> (  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  )
 <b>and </b><br/><a NAME="E4:58_1_2"/><i><font color="Green" title="E50">A67</font></i>: 
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C1</font>  st  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b2">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="Olive" title="b1">f</font> holds <br/><span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c3">C1</font>
 <b>and </b><br/><a NAME="E5:58_1_2"/><i><font color="Green" title="E51">A68</font></i>: 
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C1</font>  st <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c3">C1</font> holds <br/> ex <font color="Olive" title="b3">F</font>, <font color="Olive" title="b4">F1</font>, <font color="Olive" title="b5">F2</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> ex <font color="Olive" title="b6">t</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b3">F</font>,<font color="Olive" title="b4">F1</font> ex <font color="Olive" title="b7">t1</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font> st <br/>( <font color="Olive" title="b1">f</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="Olive" title="b3">F</font>,<font color="Olive" title="b4">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b6">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">g</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="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b7">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c3">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <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="Olive" title="b3">F</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b7">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Olive" title="b6">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>and </b><br/><a NAME="E6:58_1_2"/><i><font color="Green" title="E52">A69</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c3">C1</font><br/> for <font color="Olive" title="b2">F</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>  st <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> holds <br/> <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Olive" title="b1">a</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="Olive" title="b2">F</font>,<font color="Olive" title="b2">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Olive" title="b2">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E7:58_1_2"/><i><font color="Green" title="E53">A70</font></i>: 
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c4">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>
 <b>and </b><br/><a NAME="E8:58_1_2"/><i><font color="Green" title="E54">A71</font></i>: 
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c4">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>
 <b>and </b><br/><a NAME="E9:58_1_2"/><i><font color="Green" title="E55">A72</font></i>: 
for <font color="Olive" title="b1">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">C2</font> holds <br/> (  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  )
 <b>and </b><br/><a NAME="E10:58_1_2"/><i><font color="Green" title="E56">A73</font></i>: 
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">C2</font>  st  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b2">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="Olive" title="b1">f</font> holds <br/><span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c4">C2</font>
 <b>and </b><br/><a NAME="E11:58_1_2"/><i><font color="Green" title="E57">A74</font></i>: 
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">C2</font>  st <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c4">C2</font> holds <br/> ex <font color="Olive" title="b3">F</font>, <font color="Olive" title="b4">F1</font>, <font color="Olive" title="b5">F2</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> ex <font color="Olive" title="b6">t</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b3">F</font>,<font color="Olive" title="b4">F1</font> ex <font color="Olive" title="b7">t1</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font> st <br/>( <font color="Olive" title="b1">f</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="Olive" title="b3">F</font>,<font color="Olive" title="b4">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b6">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">g</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="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b7">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c4">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <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="Olive" title="b3">F</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b7">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Olive" title="b6">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>and </b><br/><a NAME="E12:58_1_2"/><i><font color="Green" title="E58">A75</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c4">C2</font><br/> for <font color="Olive" title="b2">F</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>  st <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> holds <br/> <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Olive" title="b1">a</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="Olive" title="b2">F</font>,<font color="Olive" title="b2">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Olive" title="b2">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_2_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:58_1_2_1"/>
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">C1</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="c4">C2</font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E2:58_1_2"><i><font color="Green" title="E48">A65</font></i></a>, <a class="txt" href="nattra_1.html#E8:58_1_2"><i><font color="Green" title="E54">A71</font></i></a></i>;<br/><b>thus </b><a NAME="E2:58_1_2_1"/>
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">C1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c4">C2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_2_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:58_1_2_1_1"/>
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">C1</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="c4">C2</font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E2:58_1_2"><i><font color="Green" title="E48">A65</font></i></a>, <a class="txt" href="nattra_1.html#E8:58_1_2"><i><font color="Green" title="E54">A71</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_2.html#D8" title="FUNCT_2:def.8">FUNCT_2:def 8</a><br/>

<b>let </b><font color="Maroon" title="c5">a</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C1</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c6">b</font> = <font color="Maroon" title="c5">a</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">C2</font> <b>by </b><i><a class="txt" href="nattra_1.html#E2:58_1_2"><i><font color="Green" title="E48">A65</font></i></a>, <a class="txt" href="nattra_1.html#E8:58_1_2"><i><font color="Green" title="E54">A71</font></i></a></i>;<br/>
<b>thus </b>the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font> = 
 <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c5">a</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="nattra_1.html#E3:58_1_2"><i><font color="Green" title="E49">A66</font></i></a></i>
<br/>.= 
 <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c6">b</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E9:58_1_2"><i><font color="Green" title="E55">A72</font></i></a></i>
<br/>.= 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c4">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font>

;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:58_1_2_1"/>
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">C1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c4">C2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_2_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:58_1_2_1_2"/>
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">C1</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="c4">C2</font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E2:58_1_2"><i><font color="Green" title="E48">A65</font></i></a>, <a class="txt" href="nattra_1.html#E8:58_1_2"><i><font color="Green" title="E54">A71</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_2.html#D8" title="FUNCT_2:def.8">FUNCT_2:def 8</a><br/>

<b>let </b><font color="Maroon" title="c5">a</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C1</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c6">b</font> = <font color="Maroon" title="c5">a</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">C2</font> <b>by </b><i><a class="txt" href="nattra_1.html#E2:58_1_2"><i><font color="Green" title="E48">A65</font></i></a>, <a class="txt" href="nattra_1.html#E8:58_1_2"><i><font color="Green" title="E54">A71</font></i></a></i>;<br/>
<b>thus </b>the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font> = 
 <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">a</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="nattra_1.html#E3:58_1_2"><i><font color="Green" title="E49">A66</font></i></a></i>
<br/>.= 
 <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c6">b</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E9:58_1_2"><i><font color="Green" title="E55">A72</font></i></a></i>
<br/>.= 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c4">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font>

;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_2_1_3"><b>now </b></a><div class="add"><a NAME="E1:58_1_2_1_3"/><i><font color="Green" title="E59">A76</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c3">C1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">C1</font>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">C1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">S1</font> =  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c3">C1</font>, <font color="Maroon" title="c6">S2</font> =  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c4">C2</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">C1</font>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">C1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <b>by </b><i><a class="txt" href="nattra_1.html#E2:58_1_2"><i><font color="Green" title="E48">A65</font></i></a>, <a class="txt" href="nattra_1.html#E8:58_1_2"><i><font color="Green" title="E54">A71</font></i></a>, <a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_2_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">C1</font>,the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">C1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/><a NAME="E1:58_1_2_1_3_1"/><i><font color="Green" title="E60">A77</font></i>: 
<font color="Maroon" title="c7">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">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">f1</font> = <font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> , <font color="Maroon" title="c9">g1</font> = <font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C1</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">f2</font> = <font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> , <font color="Maroon" title="c11">g2</font> = <font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">C2</font> <b>by </b><i><a class="txt" href="nattra_1.html#E2:58_1_2"><i><font color="Green" title="E48">A65</font></i></a>, <a class="txt" href="nattra_1.html#E8:58_1_2"><i><font color="Green" title="E54">A71</font></i></a>, <a class="txt" href="nattra_1.html#E1:58_1_2_1_3_1"><i><font color="Green" title="E60">A77</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><div><i><font color="Green" title="E61">A78</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_2_1_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:58_1_2_1_3_1_1"/>
<span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c9">g1</font>,<font color="Maroon" title="c8">f1</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S1</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c12">F</font>, <font color="Maroon" title="c13">F1</font>, <font color="Maroon" title="c14">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c15">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c12">F</font>,<font color="Maroon" title="c13">F1</font>, <font color="Maroon" title="c16">t1</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c13">F1</font>,<font color="Maroon" title="c14">F2</font><b> such that </b><br/><a NAME="E3:58_1_2_1_3_1_1"/><i><font color="Green" title="E62">A79</font></i>: 
( <font color="Maroon" title="c8">f1</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="c12">F</font>,<font color="Maroon" title="c13">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c15">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c9">g1</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="c13">F1</font>,<font color="Maroon" title="c14">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c16">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>and </b><br/><a NAME="E4:58_1_2_1_3_1_1"/>
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c3">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c9">g1</font>,<font color="Maroon" title="c8">f1</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <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="c12">F</font>,<font color="Maroon" title="c14">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c15">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="nattra_1.html#E5:58_1_2"><i><font color="Green" title="E51">A68</font></i></a></i>;<br/> <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c11">g2</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">g2</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="nattra_1.html#E9:58_1_2"><i><font color="Green" title="E55">A72</font></i></a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F1</font>,<font color="Maroon" title="c14">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="nattra_1.html#E3:58_1_2_1_3_1_1"><i><font color="Green" title="E62">A79</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon" title="c13">F1</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">F</font>,<font color="Maroon" title="c13">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="nattra_1.html#E3:58_1_2_1_3_1_1"><i><font color="Green" title="E62">A79</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c10">f2</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E9:58_1_2"><i><font color="Green" title="E55">A72</font></i></a></i>
;<br/><b>hence </b><a NAME="E6:58_1_2_1_3_1_1"/>
<span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c11">g2</font>,<font color="Maroon" title="c10">f2</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S2</font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E10:58_1_2"><i><font color="Green" title="E56">A73</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_2_1_3_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:58_1_2_1_3_1_2"/>
<span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c11">g2</font>,<font color="Maroon" title="c10">f2</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S2</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c12">F</font>, <font color="Maroon" title="c13">F1</font>, <font color="Maroon" title="c14">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c15">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c12">F</font>,<font color="Maroon" title="c13">F1</font>, <font color="Maroon" title="c16">t1</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c13">F1</font>,<font color="Maroon" title="c14">F2</font><b> such that </b><br/><a NAME="E3:58_1_2_1_3_1_2"/><i><font color="Green" title="E62">A80</font></i>: 
( <font color="Maroon" title="c10">f2</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="c12">F</font>,<font color="Maroon" title="c13">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c15">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c11">g2</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="c13">F1</font>,<font color="Maroon" title="c14">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c16">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>and </b><br/><a NAME="E4:58_1_2_1_3_1_2"/>
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c4">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c11">g2</font>,<font color="Maroon" title="c10">f2</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <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="c12">F</font>,<font color="Maroon" title="c14">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c15">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="nattra_1.html#E11:58_1_2"><i><font color="Green" title="E57">A74</font></i></a></i>;<br/> <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c9">g1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">g1</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="nattra_1.html#E3:58_1_2"><i><font color="Green" title="E49">A66</font></i></a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F1</font>,<font color="Maroon" title="c14">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="nattra_1.html#E3:58_1_2_1_3_1_2"><i><font color="Green" title="E62">A80</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon" title="c13">F1</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">F</font>,<font color="Maroon" title="c13">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="nattra_1.html#E3:58_1_2_1_3_1_2"><i><font color="Green" title="E62">A80</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c8">f1</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E3:58_1_2"><i><font color="Green" title="E49">A66</font></i></a></i>
;<br/><b>hence </b><a NAME="E6:58_1_2_1_3_1_2"/>
<span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c9">g1</font>,<font color="Maroon" title="c8">f1</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S1</font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E4:58_1_2"><i><font color="Green" title="E50">A67</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:58_1_2_1_3_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">S1</font> iff <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S2</font> )
 <b>by </b><i><a class="txt" href="nattra_1.html#E4:58_1_2_1_3_1"><i><font color="Green" title="E61">A78</font></i></a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:58_1_2_1_3"/><i><font color="Green" title="E60">A81</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c3">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c4">C2</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T8" title="SUBSET_1:th.8">SUBSET_1:8</a></i>;<br/><b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E5:58_1_2_1_3"/><i><font color="Green" title="E61">A82</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c3">C1</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> , <font color="Maroon" title="c9">g</font> = <font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C1</font> <b>by </b><i><a class="txt" href="nattra_1.html#E1:58_1_2_1_3"><i><font color="Green" title="E59">A76</font></i></a>, <a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></i>;<br/><a NAME="E7:58_1_2_1_3"/><i><font color="Green" title="E62">A83</font></i>: 
<span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E1:58_1_2_1_3"><i><font color="Green" title="E59">A76</font></i></a>, <a class="txt" href="nattra_1.html#E5:58_1_2_1_3"><i><font color="Green" title="E61">A82</font></i></a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">F</font>, <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 <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</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="c10">F</font>,<font color="Maroon" title="c11">F1</font>, <font color="Maroon" title="c14">t1</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="E9:58_1_2_1_3"/><i><font color="Green" title="E63">A84</font></i>: 
( <font color="Maroon" title="c8">f</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="c10">F</font>,<font color="Maroon" title="c11">F1</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="c9">g</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="c14">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c3">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <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="c10">F</font>,<font color="Maroon" title="c12">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c13">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="nattra_1.html#E5:58_1_2"><i><font color="Green" title="E51">A68</font></i></a>, <a class="txt" href="nattra_1.html#E5:58_1_2_1_3"><i><font color="Green" title="E61">A82</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c15">F'</font>, <font color="Maroon" title="c16">F1'</font>, <font color="Maroon" title="c17">F2'</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c18">t'</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c15">F'</font>,<font color="Maroon" title="c16">F1'</font>, <font color="Maroon" title="c19">t1'</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c16">F1'</font>,<font color="Maroon" title="c17">F2'</font><b> such that </b><br/><a NAME="E11:58_1_2_1_3"/><i><font color="Green" title="E64">A85</font></i>: 
( <font color="Maroon" title="c8">f</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="c15">F'</font>,<font color="Maroon" title="c16">F1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c18">t'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c9">g</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="c16">F1'</font>,<font color="Maroon" title="c17">F2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c19">t1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c4">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <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="c15">F'</font>,<font color="Maroon" title="c17">F2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">t1'</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c18">t'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="nattra_1.html#E2:58_1_2"><i><font color="Green" title="E48">A65</font></i></a>, <a class="txt" href="nattra_1.html#E8:58_1_2"><i><font color="Green" title="E54">A71</font></i></a>, <a class="txt" href="nattra_1.html#E11:58_1_2"><i><font color="Green" title="E57">A74</font></i></a>, <a class="txt" href="nattra_1.html#E4:58_1_2_1_3"><i><font color="Green" title="E60">A81</font></i></a>, <a class="txt" href="nattra_1.html#E5:58_1_2_1_3"><i><font color="Green" title="E61">A82</font></i></a>, <a class="txt" href="nattra_1.html#E7:58_1_2_1_3"><i><font color="Green" title="E62">A83</font></i></a></i>;<br/><a NAME="E12:58_1_2_1_3"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">F</font>,<font color="Maroon" title="c11">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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"><font color="Maroon" title="c15">F'</font>,<font color="Maroon" title="c16">F1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <span class="p1"><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> <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"><font color="Maroon" title="c16">F1'</font>,<font color="Maroon" title="c17">F2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="nattra_1.html#E9:58_1_2_1_3"><i><font color="Green" title="E63">A84</font></i></a>, <a class="txt" href="nattra_1.html#E11:58_1_2_1_3"><i><font color="Green" title="E64">A85</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E13:58_1_2_1_3"/><b>then </b>
( <font color="Maroon" title="c10">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c15">F'</font> &amp; <font color="Maroon" title="c11">F1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c16">F1'</font> &amp; <font color="Maroon" title="c12">F2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c17">F2'</font> &amp; <font color="Maroon" title="c13">t</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c18">t'</font> &amp; <font color="Maroon" title="c14">t1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c19">t1'</font> )
 <b>by </b><i><a class="txt" href="nattra_1.html#E9:58_1_2_1_3"><i><font color="Green" title="E63">A84</font></i></a>, <a class="txt" href="nattra_1.html#E11:58_1_2_1_3"><i><font color="Green" title="E64">A85</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E14:58_1_2_1_3"/>
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c3">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c4">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E7:58_1_2_1_3"><i><font color="Green" title="E62">A83</font></i></a>, <a class="txt" href="nattra_1.html#E9:58_1_2_1_3"><i><font color="Green" title="E63">A84</font></i></a>, <a class="txt" href="nattra_1.html#E11:58_1_2_1_3"><i><font color="Green" title="E64">A85</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:58_1_2_1"/>
the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c3">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <font color="Maroon" title="c4">C2</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/><b>thus </b><a NAME="E6:58_1_2_1"/>
the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c3">C1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c4">C2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_2_1_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:58_1_2_1_4"/>
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c3">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c4">C2</font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E1:58_1_2"><i><font color="Green" title="E47">A64</font></i></a>, <a class="txt" href="nattra_1.html#E7:58_1_2"><i><font color="Green" title="E53">A70</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_2.html#D8" title="FUNCT_2:def.8">FUNCT_2:def 8</a><br/>

<b>let </b><font color="Maroon" title="c5">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c3">C1</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c6">F</font> = <font color="Maroon" title="c5">a</font> as    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <b>by </b><i><a class="txt" href="nattra_1.html#E1:58_1_2"><i><font color="Green" title="E47">A64</font></i></a>, <a class="ref" href="cat_2.html#D3" title="CAT_2:def.3">CAT_2:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">b</font> = <font color="Maroon" title="c5">a</font> as   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c4">C2</font> <b>by </b><i><a class="txt" href="nattra_1.html#E1:58_1_2"><i><font color="Green" title="E47">A64</font></i></a>, <a class="txt" href="nattra_1.html#E7:58_1_2"><i><font color="Green" title="E53">A70</font></i></a></i>;<br/>
<b>thus </b>the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c3">C1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font> = 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c5">a</font>

<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c6">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c6">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="nattra_1.html#E6:58_1_2"><i><font color="Green" title="E52">A69</font></i></a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c7">b</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E12:58_1_2"><i><font color="Green" title="E58">A75</font></i></a></i>
<br/>.= 
the <a href="cat_1.html#U2" title="CAT_1:sel.2">Id</a> of <font color="Maroon" title="c4">C2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font>

;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E14:58_1_2"/>
<font color="Maroon" title="c3">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C2</font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E1:58_1_2"><i><font color="Green" title="E47">A64</font></i></a>, <a class="txt" href="nattra_1.html#E7:58_1_2"><i><font color="Green" title="E53">A70</font></i></a></i>;<br/>


</div>
