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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">G2</font> being   <a href="glib_000.html#M1" title="GLIB_000:mode.1">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/><span class="kw">let </span><font color="Maroon" title="c2">G2</font> be    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:180"/><span class="lab"><font color="Green" title="E69">A1</font></span>: 
<font color="Maroon" title="c3">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v2</font>
 <span class="kw">and </span><br/><a NAME="E2:180"/><span class="lab"><font color="Green" title="E70">A2</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">consider </span><font color="Maroon" title="c5">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:180"/><span class="lab"><font color="Green" title="E71">A3</font></span>: 
<font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.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>
 <span class="kw">and </span><br/><a NAME="E5:180"/><span class="lab"><font color="Green" title="E72">A4</font></span>: 
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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:180"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="ref" href="glib_000.html#D53" target="_self" title="GLIB_000:def.53">Def53</a></span>;<br/>
<a NAME="E6:180"/>
<font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.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#K28" title="GLIB_000:func.28">.edgesInOut()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:180"><span class="lab"><font color="Green" title="E69">A1</font></span></a>, <a class="ref" href="glib_000.html#T81" target="_self" title="GLIB_000:th.81">Th81</a></span>;<br/>
<a NAME="E7:180"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A5</font></span>: 
( <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.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#K28" title="GLIB_000:func.28">.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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:180"><span class="lab"><font color="Green" title="E71">A3</font></span></a>, <a class="ref" href="zfmisc_1.html#T39" title="ZFMISC_1:th.39">ZFMISC_1:39</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="180_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:180_1"/><span class="lab"><font color="Green" title="E74">A6</font></span>: 
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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E2:180_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:180"><span class="lab"><font color="Green" title="E69">A1</font></span></a>, <a class="txt" href="#E5:180"><span class="lab"><font color="Green" title="E72">A4</font></span></a>, <a class="txt" href="glib_000.html#E39"><span class="lab"><font color="Green" title="E22">Lm5</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:180_1"/>
<font color="Maroon" title="c4">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:180"><span class="lab"><font color="Green" title="E73">A5</font></span></a>, <a class="txt" href="#E1:180_1"><span class="lab"><font color="Green" title="E74">A6</font></span></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></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:180"/>
( <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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