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<b>let </b><font color="Maroon" title="c1">G1</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">G2</font> being   <a href="glib_000.html#M2" title="GLIB_000:mode.2">Subgraph</a> of <font color="Maroon" title="c1">G1</font><br/> for <font color="Olive" title="b2">v1</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font><br/> for <font color="Olive" title="b3">v2</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b1">G2</font>  st <font color="Olive" title="b2">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v2</font> &amp; <font color="Olive" title="b2">v1</font> is <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> &amp; not <font color="Olive" title="b3">v2</font> is <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> holds <br/><font color="Olive" title="b3">v2</font> is <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a></span><br/><b>let </b><font color="Maroon" title="c2">G2</font> be    <a href="glib_000.html#M2" title="GLIB_000:mode.2">Subgraph</a> of <font color="Maroon" title="c1">G1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">v1</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font><br/> for <font color="Olive" title="b2">v2</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b1">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v2</font> &amp; <font color="Olive" title="b1">v1</font> is <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> &amp; not <font color="Olive" title="b2">v2</font> is <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> holds <br/><font color="Olive" title="b2">v2</font> is <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a></span><br/><b>let </b><font color="Maroon" title="c3">v1</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">v2</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Maroon" title="c3">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v2</font> &amp; <font color="Maroon" title="c3">v1</font> is <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> &amp; not <font color="Olive" title="b1">v2</font> is <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> holds <br/><font color="Olive" title="b1">v2</font> is <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a></span><br/><b>let </b><font color="Maroon" title="c4">v2</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v2</font> &amp; <font color="Maroon" title="c3">v1</font> is <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> &amp; not <font color="Maroon" title="c4">v2</font> is <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> implies <font color="Maroon" title="c4">v2</font> is <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a> )</span><br/>







<b>assume </b><a NAME="E1:176"/><i><font color="Green" title="E70">A1</font></i>: 
( <font color="Maroon" title="c3">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v2</font> &amp; <font color="Maroon" title="c3">v1</font> is <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">v2</font> is <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> or <font color="Maroon" title="c4">v2</font> is <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a> )</span><br/>

<a NAME="E2:176"/><b>then </b><i><font color="Green" title="E71">A2</font></i>: 
<font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a> 
 
<b>by </b><i><a class="ref" href="glib_000.html#T81" target="_self" title="GLIB_000:th.81">Th81</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:176"/><i><font color="Green" title="E72">A3</font></i>: 
( <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; not <font color="Maroon" title="c5">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c1">G1</font> )
 <b>by </b><i><a class="txt" href="glib_000.html#E1:176"><i><font color="Green" title="E70">A1</font></i></a>, <a class="ref" href="glib_000.html#D53" target="_self" title="GLIB_000:def.53">Def53</a></i>;<br/>
<a NAME="E5:176"/><i><font color="Green" title="E73">A4</font></i>: 
( <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="glib_000.html#E2:176"><i><font color="Green" title="E71">A2</font></i></a>, <a class="txt" href="glib_000.html#E4:176"><i><font color="Green" title="E72">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#T39" title="ZFMISC_1:th.39">ZFMISC_1:39</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="176_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:176_1"/><i><font color="Green" title="E74">A5</font></i>: 
not <font color="Maroon" title="c4">v2</font> is <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">v2</font> is <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a></span><br/><a NAME="E2:176_1"/>
not <font color="Maroon" title="c5">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">v2</font>,<font color="Maroon" title="c4">v2</font>,<font color="Maroon" title="c2">G2</font>
 <b>by </b><i><a class="txt" href="glib_000.html#E1:176"><i><font color="Green" title="E70">A1</font></i></a>, <a class="txt" href="glib_000.html#E4:176"><i><font color="Green" title="E72">A3</font></i></a>, <a class="txt" href="glib_000.html#E39"><i><font color="Green" title="E23">Lm5</font></i></a></i>;<br/><b>hence </b><a NAME="E3:176_1"/>
<font color="Maroon" title="c4">v2</font> is <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>
 <b>by </b><i><a class="txt" href="glib_000.html#E5:176"><i><font color="Green" title="E73">A4</font></i></a>, <a class="txt" href="glib_000.html#E1:176_1"><i><font color="Green" title="E74">A5</font></i></a>, <a class="ref" href="glib_000.html#D51" target="_self" title="GLIB_000:def.51">Def51</a>, <a class="ref" href="glib_000.html#D53" target="_self" title="GLIB_000:def.53">Def53</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:176"/>
( <font color="Maroon" title="c4">v2</font> is <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> or <font color="Maroon" title="c4">v2</font> is <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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