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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#K1" title="GLIB_014:func.1">the_Vertices_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> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<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><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="115_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>  <a href="glib_014.html#K1" title="GLIB_014:func.1">the_Vertices_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> implies <font color="Olive" title="b1">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> ) &amp; ( <font color="Olive" title="b1">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> implies <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_014.html#K1" title="GLIB_014:func.1">the_Vertices_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="glib_014.html#K1" title="GLIB_014:func.1">the_Vertices_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> implies <font color="Maroon" title="c2">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> ) &amp; ( <font color="Maroon" title="c2">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> implies <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_014.html#K1" title="GLIB_014:func.1">the_Vertices_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/><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>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font> implies <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_014.html#K1" title="GLIB_014:func.1">the_Vertices_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>
<div class="add"><span class="kw">assume </span><a NAME="E1:115_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_014.html#K1" title="GLIB_014:func.1">the_Vertices_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 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>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">H</font> being   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a><span class="kw"> such that </span><br/><a NAME="E3:115_1_1"/><span class="lab"><font color="Green" title="E62">A1</font></span>: 
( <font color="Maroon" title="c3">H</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>  &amp; <font color="Maroon" title="c2">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="c3">H</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_014.html#D14" onmouseover="rs('glib_014/D14')" onmouseout="rh()">GLIB_014:def 14</a></span>;<br/><a NAME="E4:115_1_1"/>
<font color="Maroon" title="c3">H</font> is   <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:115_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">hence </span><a NAME="E5:115_1_1"/>
<font color="Maroon" title="c2">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="#E3:115_1_1"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="ref" href="glib_000.html#D33" onmouseover="rs('glib_000/D33')" onmouseout="rh()">GLIB_000:def 33</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="E2:115_1"/><span class="lab"><font color="Green" title="E62">A2</font></span>: 
<font color="Maroon" title="c2">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>
 ; <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#K1" title="GLIB_014:func.1">the_Vertices_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">set </span><font color="Maroon" title="c3">H</font> =  the   <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</font>;<br/><a NAME="E3:115_1"/>
(  the   <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</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>  &amp;  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</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="ref" href="glenum00.html#T60" target="_self" onmouseover="rs('glenum00/T60')" onmouseout="rh()">Th60</a>, <a class="ref" href="glib_000.html#D33" onmouseover="rs('glib_000/D33')" onmouseout="rh()">GLIB_000:def 33</a></span>;<br/><span class="kw">hence </span><a NAME="E4:115_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_014.html#K1" title="GLIB_014:func.1">the_Vertices_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="#E2:115_1"><span class="lab"><font color="Green" title="E62">A2</font></span></a>, <a class="ref" href="glib_014.html#D14" onmouseover="rs('glib_014/D14')" onmouseout="rh()">GLIB_014:def 14</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="E2:115"/>
 <a href="glib_014.html#K1" title="GLIB_014:func.1">the_Vertices_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> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<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><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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