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<b>let </b><font color="Maroon" title="c3">G1</font>, <font color="Maroon" title="c4">G2</font>, <font color="Maroon" title="c5">G3</font> be   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a>;<br/>





<b>assume </b><a NAME="E1:53"/>
( <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">G2</font> &amp; <font color="Maroon" title="c4">G2</font> <a href="graph_1.html#R4" title="GRAPH_1:pred.4">c=</a> <font color="Maroon" title="c5">G3</font> )
 ;<br/>

<a NAME="E2:53"/><b>then </b><i><font color="Green" title="E32">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">G2</font> &amp; <font color="Maroon" title="c4">G2</font> is    <a href="graph_1.html#M3" title="GRAPH_1:mode.3">Subgraph</a> of <font color="Maroon" title="c5">G3</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="E3:53"/><b>then </b>
( the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</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#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c4">G2</font> &amp; the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c4">G2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c5">G3</font> )
 
<b>by </b><i><a class="ref" href="graph_1.html#D17" target="_self" title="GRAPH_1:def.17">Def17</a></i>;<br/>
<a NAME="E4:53"/><b>then </b><i><font color="Green" title="E33">A2</font></i>: 
the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</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#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c5">G3</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E5:53"/><i><font color="Green" title="E34">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">G2</font> &amp; the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c4">G2</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="c5">G3</font> )
 
<b>by </b><i><a class="txt" href="graph_1.html#E2:53"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="graph_1.html#D17" target="_self" title="GRAPH_1:def.17">Def17</a></i>;<br/>
<a NAME="E6:53"/><b>then </b><i><font color="Green" title="E35">A4</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="c5">G3</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E7:53"/>
for <font color="Olive" title="b1">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">v</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> holds <br/>( 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">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c5">G3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font> &amp; 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">v</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="c5">G3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font> &amp; the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c5">G3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c3">G1</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c5">G3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c3">G1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:53_1"/><i><font color="Green" title="E36">A5</font></i>: 
<font color="Maroon" title="c6">v</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>
 ;<br/>

<b>hence </b><i><font color="Green" title="E37">A6</font></i>: 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="c6">v</font> = 
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c4">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E2:53"><i><font color="Green" title="E32">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="c5">G3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E2:53"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="graph_1.html#E5:53"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="graph_1.html#E1:53_1"><i><font color="Green" title="E36">A5</font></i></a>, <a class="ref" href="graph_1.html#D17" target="_self" title="GRAPH_1:def.17">Def17</a></i>
;<br/><br/>

<b>thus </b><i><font color="Green" title="E38">A7</font></i>: 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="c6">v</font> = 
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c4">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E2:53"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="graph_1.html#E1:53_1"><i><font color="Green" title="E36">A5</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="c5">G3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font>
<b>by </b><i><a class="txt" href="graph_1.html#E2:53"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="graph_1.html#E5:53"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="graph_1.html#E1:53_1"><i><font color="Green" title="E36">A5</font></i></a>, <a class="ref" href="graph_1.html#D17" target="_self" title="GRAPH_1:def.17">Def17</a></i>
;<br/>

<a NAME="E4:53_1"/><i><font color="Green" title="E39">A8</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</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#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c3">G1</font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E5:53_1"/>
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">G1</font>
 
<b>by </b><i><a class="txt" href="graph_1.html#E1:53_1"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="graph_1.html#E3"><i><font color="Green" title="E2">Lm1</font></i></a></i>;<br/>
<a NAME="E6:53_1"/><b>then </b>
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="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">G1</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:53_1"/>
the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c5">G3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c3">G1</font>
 <b>by </b><i><a class="txt" href="graph_1.html#E2:53_1"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="graph_1.html#E4:53_1"><i><font color="Green" title="E39">A8</font></i></a></i>;<br/>

<a NAME="E8:53_1"/><i><font color="Green" title="E40">A9</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</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="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c3">G1</font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E9:53_1"/>
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">G1</font>
 
<b>by </b><i><a class="txt" href="graph_1.html#E1:53_1"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="graph_1.html#E3"><i><font color="Green" title="E2">Lm1</font></i></a></i>;<br/>
<a NAME="E10:53_1"/><b>then </b>
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="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">G1</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E11:53_1"/>
the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c5">G3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Vertices</a> of <font color="Maroon" title="c3">G1</font>
 <b>by </b><i><a class="txt" href="graph_1.html#E3:53_1"><i><font color="Green" title="E38">A7</font></i></a>, <a class="txt" href="graph_1.html#E8:53_1"><i><font color="Green" title="E40">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:53"/><b>then </b>
<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="c5">G3</font>
 
<b>by </b><i><a class="txt" href="graph_1.html#E4:53"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="graph_1.html#E6:53"><i><font color="Green" title="E35">A4</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="E9:53"/>
<font color="Maroon" title="c3">G1</font> <a href="graph_1.html#R4" title="GRAPH_1:pred.4">c=</a> <font color="Maroon" title="c5">G3</font>
 <b>by </b><i><a class="ref" href="graph_1.html#D23" target="_self" title="GRAPH_1:def.23">Def23</a></i>;<br/>


</div>
