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<span class="kw">let </span><font color="Maroon" title="c1">G1</font> be   <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>   <a href="glib_003.html#V7" title="GLIB_003:attr.7">real-weighted</a>  <a href="glib_003.html#NM1" title="GLIB_003:NM.1">WGraph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/>  for <font color="Olive" title="b2">G2</font> being   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">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#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> holds  <font color="Olive" title="b2">G2</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G2</font> being   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">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#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> holds  <font color="Olive" title="b1">G2</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> </span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">V</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">G2</font> be   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">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#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">G2</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">V</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_004.html#T30" target="_self" title="GLIB_004:th.30">Th30</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">E</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">E</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K16" title="GLIB_004:func.16">PRIM:CompSeq</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="glib_004.html#K15" title="GLIB_004:func.15">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Maroon" title="c5">V</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_004.html#T30" target="_self" title="GLIB_004:th.30">Th30</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">G3</font> =  the    <a href="glib_000.html#M2" title="GLIB_000:mode.2">inducedSubgraph</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c5">V</font>,<font color="Maroon" title="c7">E</font>;<br/>
<a NAME="E3:66"/><span class="lab"><font color="Green" title="E45">A1</font></span>: 
(  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the    <a href="glib_000.html#M2" title="GLIB_000:mode.2">inducedSubgraph</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c5">V</font>,<font color="Maroon" title="c7">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">V</font> &amp;  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c4">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">V</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D37" title="GLIB_000:def.37">GLIB_000:def 37</a></span>;<br/>
<a NAME="E4:66"/>
(  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a>  the    <a href="glib_000.html#M2" title="GLIB_000:mode.2">inducedSubgraph</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c5">V</font>,<font color="Maroon" title="c7">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">E</font> &amp;  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c4">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Maroon" title="c5">V</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D37" title="GLIB_000:def.37">GLIB_000:def 37</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">G3</font> =  the    <a href="glib_000.html#M2" title="GLIB_000:mode.2">inducedSubgraph</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c5">V</font>,<font color="Maroon" title="c7">E</font> as    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c4">G2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:66"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="ref" href="glib_000.html#T44" title="GLIB_000:th.44">GLIB_000:44</a></span>;<br/>
<a NAME="E6:66"/><span class="lab"><font color="Green" title="E46">A2</font></span>: 
<font color="Maroon" title="c9">G3</font> is  <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:66"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="ref" href="glib_000.html#D33" title="GLIB_000:def.33">GLIB_000:def 33</a></span>;<br/>
<a NAME="E7:66"/>
<font color="Maroon" title="c9">G3</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_004.html#T31" target="_self" title="GLIB_004:th.31">Th31</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:66"/>
<font color="Maroon" title="c4">G2</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:66"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="ref" href="glib_002.html#T23" title="GLIB_002:th.23">GLIB_002:23</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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