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<b>let </b><font color="Maroon" title="c2">IT1</font>, <font color="Maroon" title="c3">IT2</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>; <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">e1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">IT1</font> iff ( <font color="Olive" title="b1">e1</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c1">G</font> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> </span>)</span>,<font color="Maroon" title="c1">G</font> &amp; ( for <font color="Olive" title="b2">e2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">e2</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c1">G</font> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> </span>)</span>,<font color="Maroon" title="c1">G</font> holds <br/><span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">e1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">e2</font> ) ) ) ) &amp; ( for <font color="Olive" title="b1">e1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT2</font> iff ( <font color="Olive" title="b1">e1</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c1">G</font> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> </span>)</span>,<font color="Maroon" title="c1">G</font> &amp; ( for <font color="Olive" title="b2">e2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">e2</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c1">G</font> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> </span>)</span>,<font color="Maroon" title="c1">G</font> holds <br/><span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">e1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">e2</font> ) ) ) ) implies <font color="Maroon" title="c2">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">IT2</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:56_1_2"/><i><font color="Green" title="E35">A3</font></i>: 
for <font color="Olive" title="b1">e1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">IT1</font> iff ( <font color="Olive" title="b1">e1</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c1">G</font> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> </span>)</span>,<font color="Maroon" title="c1">G</font> &amp; ( for <font color="Olive" title="b2">e2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">e2</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c1">G</font> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> </span>)</span>,<font color="Maroon" title="c1">G</font> holds <br/><span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">e1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">e2</font> ) ) )
 <b>and </b><br/><a NAME="E2:56_1_2"/><i><font color="Green" title="E36">A4</font></i>: 
for <font color="Olive" title="b1">e1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT2</font> iff ( <font color="Olive" title="b1">e1</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c1">G</font> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> </span>)</span>,<font color="Maroon" title="c1">G</font> &amp; ( for <font color="Olive" title="b2">e2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">e2</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c1">G</font> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> </span>)</span>,<font color="Maroon" title="c1">G</font> holds <br/><span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">e1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">e2</font> ) ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">IT2</font></span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( $1 <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c1">G</font> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> </span>)</span>,<font color="Maroon" title="c1">G</font> &amp; ( for <font color="Olive" title="b1">e2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e2</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c1">G</font> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> </span>)</span>,<font color="Maroon" title="c1">G</font> holds <br/><span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">e2</font> ) );<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">e1</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="c4">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">IT1</font> implies <font color="Maroon" title="c4">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT2</font> ) &amp; ( <font color="Maroon" title="c4">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT2</font> implies <font color="Maroon" title="c4">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">IT1</font> ) )</span><br/><div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT2</font> implies <font color="Maroon" title="c4">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">IT1</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:56_1_2_1_1"/>
<font color="Maroon" title="c4">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">IT1</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">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT2</font></span><br/><a NAME="E2:56_1_2_1_1"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">e1</font>]
 <b>by </b><i><a class="txt" href="glib_004.html#E1:56_1_2"><i><font color="Green" title="E35">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E3:56_1_2_1_1"/>
<font color="Maroon" title="c4">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT2</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E2:56_1_2"><i><font color="Green" title="E36">A4</font></i></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>assume </b><a NAME="E2:56_1_2_1"/>
<font color="Maroon" title="c4">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT2</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">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">IT1</font></span><br/><a NAME="E3:56_1_2_1"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">e1</font>]
 <b>by </b><i><a class="txt" href="glib_004.html#E2:56_1_2"><i><font color="Green" title="E36">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E4:56_1_2_1"/>
<font color="Maroon" title="c4">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">IT1</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:56_1_2"><i><font color="Green" title="E35">A3</font></i></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="E4:56_1_2"/>
<font color="Maroon" title="c2">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">IT2</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</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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