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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> holds <br/>( <font color="Olive" title="b3">v2</font> <a href="glib_000.html#K37" title="GLIB_000:func.37">.inNeighbors()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">v1</font> <a href="glib_000.html#K37" title="GLIB_000:func.37">.inNeighbors()</a>  &amp; <font color="Olive" title="b3">v2</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.outNeighbors()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">v1</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.outNeighbors()</a>  &amp; <font color="Olive" title="b3">v2</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.allNeighbors()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">v1</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.allNeighbors()</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> holds <br/>( <font color="Olive" title="b2">v2</font> <a href="glib_000.html#K37" title="GLIB_000:func.37">.inNeighbors()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">v1</font> <a href="glib_000.html#K37" title="GLIB_000:func.37">.inNeighbors()</a>  &amp; <font color="Olive" title="b2">v2</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.outNeighbors()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">v1</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.outNeighbors()</a>  &amp; <font color="Olive" title="b2">v2</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.allNeighbors()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">v1</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.allNeighbors()</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> holds <br/>( <font color="Olive" title="b1">v2</font> <a href="glib_000.html#K37" title="GLIB_000:func.37">.inNeighbors()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K37" title="GLIB_000:func.37">.inNeighbors()</a>  &amp; <font color="Olive" title="b1">v2</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.outNeighbors()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.outNeighbors()</a>  &amp; <font color="Olive" title="b1">v2</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.allNeighbors()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.allNeighbors()</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> implies ( <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K37" title="GLIB_000:func.37">.inNeighbors()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K37" title="GLIB_000:func.37">.inNeighbors()</a>  &amp; <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.outNeighbors()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.outNeighbors()</a>  &amp; <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.allNeighbors()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.allNeighbors()</a>  ) )</span><br/>







<b>assume </b><a NAME="E1:174"/><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>
 ; <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> <a href="glib_000.html#K37" title="GLIB_000:func.37">.inNeighbors()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K37" title="GLIB_000:func.37">.inNeighbors()</a>  &amp; <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.outNeighbors()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.outNeighbors()</a>  &amp; <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.allNeighbors()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.allNeighbors()</a>  )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="174_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K37" title="GLIB_000:func.37">.inNeighbors()</a>  implies <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K37" title="GLIB_000:func.37">.inNeighbors()</a>  )</span><br/><b>assume </b><a NAME="E1:174_1"/>
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K37" title="GLIB_000:func.37">.inNeighbors()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K37" title="GLIB_000:func.37">.inNeighbors()</a> </span><br/><b>then consider </b><font color="Maroon" title="c6">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:174_1"/><i><font color="Green" title="E71">A2</font></i>: 
<font color="Maroon" title="c6">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c4">v2</font>,<font color="Maroon" title="c2">G2</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T72" target="_self" title="GLIB_000:th.72">Th72</a></i>;<br/><a NAME="E4:174_1"/>
<font color="Maroon" title="c6">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">v</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#E1:174"><i><font color="Green" title="E70">A1</font></i></a>, <a class="txt" href="glib_000.html#E3:174_1"><i><font color="Green" title="E71">A2</font></i></a>, <a class="ref" href="glib_000.html#T75" target="_self" title="GLIB_000:th.75">Th75</a></i>;<br/><b>hence </b><a NAME="E5:174_1"/>
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K37" title="GLIB_000:func.37">.inNeighbors()</a> 
 <b>by </b><i><a class="ref" href="glib_000.html#T72" target="_self" title="GLIB_000:th.72">Th72</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="E3:174"/>
<font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K37" title="GLIB_000:func.37">.inNeighbors()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K37" title="GLIB_000:func.37">.inNeighbors()</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>; <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> <a href="glib_000.html#K38" title="GLIB_000:func.38">.outNeighbors()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.outNeighbors()</a>  &amp; <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.allNeighbors()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.allNeighbors()</a>  )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="174_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.outNeighbors()</a>  implies <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.outNeighbors()</a>  )</span><br/><b>assume </b><a NAME="E1:174_2"/>
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.outNeighbors()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.outNeighbors()</a> </span><br/><b>then consider </b><font color="Maroon" title="c6">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:174_2"/><i><font color="Green" title="E71">A3</font></i>: 
<font color="Maroon" title="c6">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c4">v2</font>,<font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c2">G2</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T73" target="_self" title="GLIB_000:th.73">Th73</a></i>;<br/><a NAME="E4:174_2"/>
<font color="Maroon" title="c6">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c1">G1</font>
 <b>by </b><i><a class="txt" href="glib_000.html#E1:174"><i><font color="Green" title="E70">A1</font></i></a>, <a class="txt" href="glib_000.html#E3:174_2"><i><font color="Green" title="E71">A3</font></i></a>, <a class="ref" href="glib_000.html#T75" target="_self" title="GLIB_000:th.75">Th75</a></i>;<br/><b>hence </b><a NAME="E5:174_2"/>
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.outNeighbors()</a> 
 <b>by </b><i><a class="ref" href="glib_000.html#T73" target="_self" title="GLIB_000:th.73">Th73</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="E5:174"/>
<font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.outNeighbors()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.outNeighbors()</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>; <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> <a href="glib_000.html#K39" title="GLIB_000:func.39">.allNeighbors()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.allNeighbors()</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="174_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.allNeighbors()</a>  implies <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.allNeighbors()</a>  )</span><br/><b>assume </b><a NAME="E1:174_3"/>
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.allNeighbors()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.allNeighbors()</a> </span><br/><b>then consider </b><font color="Maroon" title="c6">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:174_3"/><i><font color="Green" title="E71">A4</font></i>: 
<font color="Maroon" title="c6">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="c5">v</font>,<font color="Maroon" title="c2">G2</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T74" target="_self" title="GLIB_000:th.74">Th74</a></i>;<br/><a NAME="E4:174_3"/>
<font color="Maroon" title="c6">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="c5">v</font>,<font color="Maroon" title="c1">G1</font>
 <b>by </b><i><a class="txt" href="glib_000.html#E1:174"><i><font color="Green" title="E70">A1</font></i></a>, <a class="txt" href="glib_000.html#E3:174_3"><i><font color="Green" title="E71">A4</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="E5:174_3"/>
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.allNeighbors()</a> 
 <b>by </b><i><a class="ref" href="glib_000.html#T74" target="_self" title="GLIB_000:th.74">Th74</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="E7:174"/>
<font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.allNeighbors()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K39" title="GLIB_000:func.39">.allNeighbors()</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>; <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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