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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>;<br/>

<b>thus </b><a NAME="E1:74"/>
( <font color="Maroon" title="c1">G</font> is   <a href="glib_002.html#NM1" title="GLIB_002:NM.1">Component</a> of <font color="Maroon" title="c1">G</font> implies <font color="Maroon" title="c1">G</font> is <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> )
 ;<br/>

<b>assume </b><a NAME="E2:74"/><i><font color="Green" title="E43">A1</font></i>: 
<font color="Maroon" title="c1">G</font> is <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>
 ;<br/>

<a NAME="E3:74"/><i><font color="Green" title="E44">A2</font></i>: 
<font color="Maroon" title="c1">G</font> is    <a href="glib_000.html#M2" title="GLIB_000:mode.2">Subgraph</a> of <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="glib_000.html#T43" title="GLIB_000:th.43">GLIB_000:43</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c2">G2</font> being  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>  <a href="glib_000.html#M2" title="GLIB_000:mode.2">Subgraph</a> of <font color="Maroon" title="c1">G</font><b> such that </b><a NAME="E2:74_1"/><i><font color="Green" title="E45">A3</font></i>: 
<font color="Maroon" title="c1">G</font> <a href="glib_000.html#R7" title="GLIB_000:pred.7">c&lt;</a> <font color="Maroon" title="c2">G2</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:74_1_1_1"/>
(  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c2">G2</font> or  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c2">G2</font> )
 </span><b>by </b><i><a class="txt" href="glib_002.html#E3:74"><i><font color="Green" title="E44">A2</font></i></a>, <a class="txt" href="glib_002.html#E2:74_1"><i><font color="Green" title="E45">A3</font></i></a>, <a class="ref" href="glib_000.html#T101" title="GLIB_000:th.101">GLIB_000:101</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_1_1"><b>suppose </b></a><a NAME="E1:74_1_1_1_1"/>
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c2">G2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:74_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_1_1_1_2"><b>suppose </b></a><a NAME="E1:74_1_1_1_2"/>
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c2">G2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:74_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:74_1"/>
contradiction
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:74"/>
<font color="Maroon" title="c1">G</font> is   <a href="glib_002.html#NM1" title="GLIB_002:NM.1">Component</a> of <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E2:74"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="glib_000.html#T43" title="GLIB_000:th.43">GLIB_000:43</a>, <a class="ref" href="glib_002.html#D7" target="_self" title="GLIB_002:def.7">Def7</a></i>;<br/>


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