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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><br/> for <font color="Olive" title="b4">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   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="b4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Olive" title="b1">G2</font> holds <br/><font color="Olive" title="b2">v1</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Olive" title="b4">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v2</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Olive" title="b4">e</font></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><br/> for <font color="Olive" title="b3">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   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="b3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c2">G2</font> holds <br/><font color="Olive" title="b1">v1</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Olive" title="b3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v2</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Olive" title="b3">e</font></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><br/> for <font color="Olive" title="b2">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   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="Olive" title="b2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c2">G2</font> holds <br/><font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Olive" title="b2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v2</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Olive" title="b2">e</font></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"> for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <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="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c2">G2</font> holds <br/><font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Olive" title="b1">e</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="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="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c2">G2</font> implies <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e</font> )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:176"/><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="E2:176"/><span class="lab"><font color="Green" title="E70">A2</font></span>: 
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <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="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e</font></span><br/>

<a NAME="E3:176"/><span class="lab"><font color="Green" title="E71">A3</font></span>: 
( <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c2">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:176"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="ref" href="glib_000.html#D34" target="_self" title="GLIB_000:def.34">Def34</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="176_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="176_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:176_1_1"/>
( <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v1</font> or ( <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v1</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">v1</font> ) or ( <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">v1</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">v1</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="176_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:176_1_1_1"/><span class="lab"><font color="Green" title="E72">A4</font></span>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v1</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="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e</font></span><br/><div class="add"><span class="kw">hence </span><font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:176"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="ref" href="glib_000.html#D43" target="_self" title="GLIB_000:def.43">Def43</a></span>
<br/>.= 
<font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:176"><span class="lab"><font color="Green" title="E69">A1</font></span></a>, <a class="txt" href="#E2:176"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="txt" href="#E3:176"><span class="lab"><font color="Green" title="E71">A3</font></span></a>, <a class="txt" href="#E1:176_1_1_1"><span class="lab"><font color="Green" title="E72">A4</font></span></a>, <a class="ref" href="glib_000.html#D43" target="_self" title="GLIB_000:def.43">Def43</a></span>
;<br/> <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="176_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:176_1_1_2"/><span class="lab"><font color="Green" title="E72">A5</font></span>: 
( <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v1</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">v1</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="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e</font></span><br/><div class="add"><span class="kw">hence </span><font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:176"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="ref" href="glib_000.html#D43" target="_self" title="GLIB_000:def.43">Def43</a></span>
<br/>.= 
<font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:176"><span class="lab"><font color="Green" title="E69">A1</font></span></a>, <a class="txt" href="#E2:176"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="txt" href="#E3:176"><span class="lab"><font color="Green" title="E71">A3</font></span></a>, <a class="txt" href="#E1:176_1_1_2"><span class="lab"><font color="Green" title="E72">A5</font></span></a>, <a class="ref" href="glib_000.html#D43" target="_self" title="GLIB_000:def.43">Def43</a></span>
;<br/> <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="176_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:176_1_1_3"/><span class="lab"><font color="Green" title="E72">A6</font></span>: 
( <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">v1</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">v1</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="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e</font></span><br/><div class="add"><span class="kw">hence </span><font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e</font> = 
<font color="Maroon" title="c4">v2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:176"><span class="lab"><font color="Green" title="E69">A1</font></span></a>, <a class="ref" href="glib_000.html#D43" target="_self" title="GLIB_000:def.43">Def43</a></span>
<br/>.= 
<font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:176"><span class="lab"><font color="Green" title="E69">A1</font></span></a>, <a class="txt" href="#E3:176"><span class="lab"><font color="Green" title="E71">A3</font></span></a>, <a class="txt" href="#E1:176_1_1_3"><span class="lab"><font color="Green" title="E72">A6</font></span></a>, <a class="ref" href="glib_000.html#D43" target="_self" title="GLIB_000:def.43">Def43</a></span>
;<br/> <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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:176"/>
<font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e</font>
 ; <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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