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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="glib_014.html#K2" title="GLIB_014:func.2">the_Edges_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <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></span><br/>

<a NAME="E1:116"/>
 <a href="glib_014.html#K2" title="GLIB_014:func.2">the_Edges_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_014.html#K2" title="GLIB_014:func.2">the_Edges_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glibpre1.html#T115" onmouseover="rs('glibpre1/T115')" onmouseout="rh()">GLIBPRE1:115</a></span>;<br/>
<a NAME="E2:116"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A1</font></span>: 
 <a href="glib_014.html#K2" title="GLIB_014:func.2">the_Edges_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T38" target="_self" onmouseover="rs('glenum00/T38')" onmouseout="rh()">Th38</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="116_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <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> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_014.html#K2" title="GLIB_014:func.2">the_Edges_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</a></span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <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> implies <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_014.html#K2" title="GLIB_014:func.2">the_Edges_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</a></span>)</span> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c3">X</font> = <font color="Maroon" title="c2">x</font> as    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" onmouseover="rs('tarski/T1')" onmouseout="rh()">TARSKI:1</a></span>;<br/><span class="kw">assume </span><a NAME="E2:116_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_014.html#K2" title="GLIB_014:func.2">the_Edges_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</a></span>)</span></span><br/><a NAME="E3:116_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font>
 ;<br/><a NAME="E4:116_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A2</font></span>: 
<font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T34" onmouseover="rs('glib_000/T34')" onmouseout="rh()">GLIB_000:34</a></span>;<br/><a NAME="E5:116_1"/>
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>
 ;<br/><a NAME="E6:116_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A3</font></span>: 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font> is   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">G</font></span>)</span>
 ;<br/><span class="kw">set </span><font color="Maroon" title="c4">H</font> =  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_000.html#M2" title="GLIB_000:mode.2">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>, <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">X</font>;<br/><a NAME="E7:116_1"/>
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_000.html#M2" title="GLIB_000:mode.2">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>, <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:116_1"><span class="lab"><font color="Green" title="E63">A2</font></span></a>, <a class="txt" href="#E6:116_1"><span class="lab"><font color="Green" title="E64">A3</font></span></a>, <a class="ref" href="glib_000.html#D37" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</a></span>;<br/><a NAME="E8:116_1"/><span class="kw">then </span>
 the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_000.html#M2" title="GLIB_000:mode.2">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>, <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">X</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="ref" href="glib_000.html#D33" onmouseover="rs('glib_000/D33')" onmouseout="rh()">GLIB_000:def 33</a></span>;<br/><a NAME="E9:116_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A4</font></span>: 
 the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_000.html#M2" title="GLIB_000:mode.2">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>, <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T60" target="_self" onmouseover="rs('glenum00/T60')" onmouseout="rh()">Th60</a></span>;<br/><a NAME="E10:116_1"/>
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a>  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_000.html#M2" title="GLIB_000:mode.2">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>, <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:116_1"><span class="lab"><font color="Green" title="E63">A2</font></span></a>, <a class="txt" href="#E6:116_1"><span class="lab"><font color="Green" title="E64">A3</font></span></a>, <a class="ref" href="glib_000.html#D37" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</a></span>;<br/><span class="kw">hence </span><a NAME="E11:116_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_014.html#K2" title="GLIB_014:func.2">the_Edges_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:116_1"><span class="lab"><font color="Green" title="E65">A4</font></span></a>, <a class="ref" href="glib_014.html#D15" onmouseover="rs('glib_014/D15')" onmouseout="rh()">GLIB_014:def 15</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:116"/><span class="kw">then </span>
 <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <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 href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_014.html#K2" title="GLIB_014:func.2">the_Edges_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:116"/>
 <a href="glib_014.html#K2" title="GLIB_014:func.2">the_Edges_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:116"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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