<?xml version="1.0"?>
<div class="add">

<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"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</a></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="117_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>  holds <br/> ( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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> implies <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</a> ) &amp; ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</a> implies <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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> implies <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</a> ) &amp; ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</a> implies <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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/><div><span class="kw">hereby </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#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</a> implies <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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>
<div class="add"><span class="kw">assume </span><a NAME="E1:117_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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 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#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</a></span><br/><a NAME="E2:117_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A1</font></span>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN: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="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">V</font> being   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><span class="kw"> such that </span><br/><a NAME="E4:117_1_1"/><span class="lab"><font color="Green" title="E63">A2</font></span>: 
<font color="Maroon" title="c2">x</font> is   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T45" target="_self" onmouseover="rs('glenum00/T45')" onmouseout="rh()">Th45</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c4">H</font> = <font color="Maroon" title="c2">x</font> as   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c3">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:117_1_1"><span class="lab"><font color="Green" title="E63">A2</font></span></a></span>;<br/><a NAME="E6:117_1_1"/><span class="lab"><font color="Green" title="E64">A3</font></span>: 
<font color="Maroon" title="c4">H</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="#E2:117_1_1"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="ref" href="glenum00.html#T60" target="_self" onmouseover="rs('glenum00/T60')" onmouseout="rh()">Th60</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A4</font></span>:  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font> = 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c4">H</font>
<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/>.= 
<font color="Maroon" title="c3">V</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D37" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</a></span>
;<br/> <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> = 
<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 href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c4">H</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:117_1_1"><span class="lab"><font color="Green" title="E65">A4</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="E9:117_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c1">G</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c4">H</font> &amp; <font color="Maroon" title="c1">G</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c1">G</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</a> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:117_1_1"><span class="lab"><font color="Green" title="E64">A3</font></span></a>, <a class="ref" href="glib_000.html#T128" onmouseover="rs('glib_000/T128')" onmouseout="rh()">GLIB_000:128</a>, <a class="ref" href="glib_000.html#T118" onmouseover="rs('glib_000/T118')" onmouseout="rh()">GLIB_000:118</a></span>;<br/><span class="kw">hence </span><a NAME="E10:117_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T85" onmouseover="rs('glib_000/T85')" onmouseout="rh()">GLIB_000:85</a>, <a class="ref" href="glib_009.html#T44" onmouseover="rs('glib_009/T44')" onmouseout="rh()">GLIB_009:44</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><span class="kw">assume </span><a NAME="E2:117_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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="E3:117_1"/><span class="kw">then </span>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN: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#T47" target="_self" onmouseover="rs('glenum00/T47')" onmouseout="rh()">Th47</a>, <a class="ref" href="glenum00.html#T62" target="_self" onmouseover="rs('glenum00/T62')" onmouseout="rh()">Th62</a></span>;<br/><span class="kw">hence </span><a NAME="E4:117_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</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>
<span class="kw">hence </span><a NAME="E2:117"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</a></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</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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