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<b>let </b><font color="Maroon" title="c3">G1</font> be   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a>;<br/><b>let </b><font color="Maroon" title="c4">G</font> be  <a href="graph_1.html#V5" title="GRAPH_1:attr.5">simple</a> <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a>;<br/>



<b>assume </b><a NAME="E1:63"/>
<font color="Maroon" title="c3">G1</font> <a href="graph_1.html#R4" title="GRAPH_1:pred.4">c=</a> <font color="Maroon" title="c4">G</font>
 ;<br/>

<a NAME="E2:63"/><b>then </b><i><font color="Green" title="E38">A1</font></i>: 
<font color="Maroon" title="c3">G1</font> is    <a href="graph_1.html#M3" title="GRAPH_1:mode.3">Subgraph</a> of <font color="Maroon" title="c4">G</font>
 
<b>by </b><i><a class="ref" href="graph_1.html#D23" target="_self" title="GRAPH_1:def.23">Def23</a></i>;<br/>
<a NAME="E3:63"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">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">G1</font> or  not the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E2:63_1"/><i><font color="Green" title="E39">A2</font></i>: 
( <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">G1</font> &amp; the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> )
 ;<br/>

<a NAME="E3:63_1"/><i><font color="Green" title="E40">A3</font></i>: 
the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c3">G1</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="c4">G</font>
 
<b>by </b><i><a class="txt" href="graph_1.html#E2:63"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="graph_1.html#D17" target="_self" title="GRAPH_1:def.17">Def17</a></i>;<br/>
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> = 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>
<b>by </b><i><a class="txt" href="graph_1.html#E2:63"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="graph_1.html#E2:63_1"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="graph_1.html#D17" target="_self" title="GRAPH_1:def.17">Def17</a></i>
<br/>.= 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>
<b>by </b><i><a class="txt" href="graph_1.html#E2:63"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="graph_1.html#E2:63_1"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="graph_1.html#D17" target="_self" title="GRAPH_1:def.17">Def17</a></i>
;<br/>
<b>hence </b><a NAME="E5:63_1"/>
contradiction
 <b>by </b><i><a class="txt" href="graph_1.html#E2:63_1"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="graph_1.html#E3:63_1"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="graph_1.html#D6" target="_self" title="GRAPH_1:def.6">Def6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:63"/>
<font color="Maroon" title="c3">G1</font> is <a href="graph_1.html#V5" title="GRAPH_1:attr.5">simple</a>
 <b>by </b><i><a class="ref" href="graph_1.html#D6" target="_self" title="GRAPH_1:def.6">Def6</a></i>;<br/>


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