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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="cat_5.html#V2" title="CAT_5:attr.2">categorial</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">Y</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><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_5.html#V3" title="CAT_5:attr.3">Categorial</a> <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:39"/><i><font color="Green" title="E25">A1</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> <font color="Maroon" title="c1">X</font>
 <b>and </b><br/><a NAME="E2:39"/><i><font color="Green" title="E26">A2</font></i>: 
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b3">F</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b3">F</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">C1</font> iff <font color="Olive" title="b3">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> )
 <b>and </b><br/><a NAME="E3:39"/><i><font color="Green" title="E27">A3</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> <font color="Maroon" title="c1">X</font>
 <b>and </b><br/><a NAME="E4:39"/><i><font color="Green" title="E28">A4</font></i>: 
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="cat_5.html#M1" title="CAT_5:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b3">F</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b3">F</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">C2</font> iff <font color="Olive" title="b3">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> )
 ;<br/>

<a NAME="E5:39"/>
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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<div><b>hereby </b> <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="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:39_1_1"/>
<font color="Maroon" title="c5">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="c3">C1</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c6">m</font> = <font color="Maroon" title="c5">x</font> as   <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="c7">a</font> =  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c6">m</font>, <font color="Maroon" title="c8">b</font> =  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c6">m</font> as   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a> <b>by </b><i><a class="ref" href="cat_5.html#T12" target="_self" title="CAT_5:th.12">Th12</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">f</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font><b> such that </b><br/><a NAME="E5:39_1_1"/><i><font color="Green" title="E29">A5</font></i>: 
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c9">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="cat_5.html#D6" target="_self" title="CAT_5:def.6">Def6</a></i>;<br/><a NAME="E6:39_1_1"/>
<font color="Maroon" title="c9">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="txt" href="cat_5.html#E1:39"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="cat_5.html#E2:39"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="cat_5.html#E5:39_1_1"><i><font color="Green" title="E29">A5</font></i></a></i>;<br/><a NAME="E7:39_1_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> is   <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="cat_5.html#E1:39"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="cat_5.html#E4:39"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="cat_5.html#E5:39_1_1"><i><font color="Green" title="E29">A5</font></i></a></i>;<br/><b>hence </b><a NAME="E8:39_1_1"/>
<font color="Maroon" title="c5">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="c4">C2</font>
 ;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:39_1"/>
<font color="Maroon" title="c5">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="c4">C2</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">m</font> = <font color="Maroon" title="c5">x</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">C2</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">a</font> =  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c6">m</font>, <font color="Maroon" title="c8">b</font> =  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c6">m</font> as   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a> <b>by </b><i><a class="ref" href="cat_5.html#T12" target="_self" title="CAT_5:th.12">Th12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">f</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font><b> such that </b><br/><a NAME="E6:39_1"/><i><font color="Green" title="E29">A6</font></i>: 
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c9">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="cat_5.html#D6" target="_self" title="CAT_5:def.6">Def6</a></i>;<br/>
<a NAME="E7:39_1"/>
<font color="Maroon" title="c9">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="txt" href="cat_5.html#E3:39"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="cat_5.html#E4:39"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="cat_5.html#E6:39_1"><i><font color="Green" title="E29">A6</font></i></a></i>;<br/>
<a NAME="E8:39_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> is   <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="cat_5.html#E2:39"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="cat_5.html#E3:39"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="cat_5.html#E6:39_1"><i><font color="Green" title="E29">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:39_1"/>
<font color="Maroon" title="c5">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="c3">C1</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:39"/>
<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="cat_5.html#E1:39"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="cat_5.html#E3:39"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="cat_5.html#T14" target="_self" title="CAT_5:th.14">Th14</a></i>;<br/>


</div>
