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<b>let </b><font color="Maroon">IT1</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font></span>)</span>, <font color="Maroon">IT2</font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:56_1_2"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">e1</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">e1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT1</font> iff ( <font color="Olive">e1</font> <a href="glib_000.html#R3">SJoins</a> <font color="Maroon">G</font> <a href="glib_003.html#K14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>,<font color="Maroon">G</font> &amp; ( for <font color="Olive">e2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e2</font> <a href="glib_000.html#R3">SJoins</a> <font color="Maroon">G</font> <a href="glib_003.html#K14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>,<font color="Maroon">G</font> holds <br/><span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Olive">e1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Olive">e2</font> ) ) )
 <b>and </b><br/><a NAME="E2:56_1_2"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">e1</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">e1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT2</font> iff ( <font color="Olive">e1</font> <a href="glib_000.html#R3">SJoins</a> <font color="Maroon">G</font> <a href="glib_003.html#K14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>,<font color="Maroon">G</font> &amp; ( for <font color="Olive">e2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e2</font> <a href="glib_000.html#R3">SJoins</a> <font color="Maroon">G</font> <a href="glib_003.html#K14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>,<font color="Maroon">G</font> holds <br/><span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Olive">e1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Olive">e2</font> ) ) )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="glib_000.html#R3">SJoins</a> <font color="Maroon">G</font> <a href="glib_003.html#K14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>,<font color="Maroon">G</font> &amp; ( for <font color="Olive">e2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e2</font> <a href="glib_000.html#R3">SJoins</a> <font color="Maroon">G</font> <a href="glib_003.html#K14">.labeledV()</a> ,<span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>,<font color="Maroon">G</font> holds <br/><span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Olive">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">e1</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:56_1_2_1_1"/>
<font color="Maroon">e1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT1</font>
 ;<br/><a NAME="E2:56_1_2_1_1"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">e1</font>]
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:56_1_2_1_1"/>
<font color="Maroon">e1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT2</font>
 <b>by </b><i><a class="ref" href="glib_004.html#T2" target="_self">Th2</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:56_1_2_1"/>
<font color="Maroon">e1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT2</font>
 ;<br/><a NAME="E3:56_1_2_1"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">e1</font>]
 <b>by </b><i><a class="ref" href="glib_004.html#T2" target="_self">Th2</a></i>;<br/><b>hence </b><a NAME="E4:56_1_2_1"/>
<font color="Maroon">e1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT1</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:56_1_2"/>
<font color="Maroon">IT1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT2</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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