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<b>let </b><font color="Maroon">G1</font> be  <a href="glib_000.html#V2">finite</a> <a href="glib_003.html#V7">real-weighted</a> <a href="glib_003.html#NM1">WGraph</a>;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>set </b><font color="Maroon">V</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> ;<br/>
<b>set </b><font color="Maroon">E</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a> ;<br/>
<b>let </b><font color="Maroon">G2</font> be   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">V</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G1</font></span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">E</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" target="_self">PRIM:CompSeq</a> <font color="Maroon">G1</font></span>)</span> <a href="glib_004.html#K5" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K11">.labeledE()</a>  as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="glib_000.html#K22">.edgesBetween</a> <font color="Maroon">V</font></span>)</span> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">G3</font> being    <a href="glib_000.html#M3">inducedSubgraph</a> of <font color="Maroon">G1</font>,<font color="Maroon">V</font>,<font color="Maroon">E</font>;<br/>
<a NAME="E4:70"/><i><font color="Green">E34</font></i>: 
(  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">G3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font> &amp;  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> )
 
<b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/>
<a NAME="E5:70"/><i><font color="Green">E35</font></i>: 
(  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">G2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font> &amp;  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="glib_000.html#K22">.edgesBetween</a> <font color="Maroon">V</font> )
 
<b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">G3</font> = <font color="Maroon">G3</font> as    <a href="glib_000.html#M2">Subgraph</a> of <font color="Maroon">G2</font> <b>by </b><i>, <a class="ref" href="glib_000.html#T47">GLIB_000:47</a></i>;<br/>
<a NAME="E7:70"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">G3</font> is <a href="glib_002.html#V1">connected</a>
 
<b>by </b><i/>;<br/>
<a NAME="E8:70"/>
<font color="Maroon">G3</font> is <a href="glib_000.html#V9">spanning</a>
 
<b>by </b><i>, <a class="txt" href="glib_004.html#E7:70"><i><font color="Green">E47</font></i></a>, <a class="ref" href="glib_000.html#D35">GLIB_000:def 35</a></i>;<br/>
<b>hence </b><a NAME="E9:70"/>
<font color="Maroon">G2</font> is <a href="glib_002.html#V1">connected</a>
 <b>by </b><i>, <a class="ref" href="glib_002.html#T23">GLIB_002:23</a></i>;<br/>


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