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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>  <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">  for <font color="Olive" title="b1">H</font> being   <a href="glib_009.html#NM1" title="GLIB_009:NM.1">removeLoops</a> of <font color="Maroon" title="c1">G</font> holds  <font color="Olive" title="b1">H</font> is    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">H</font> be   <a href="glib_009.html#NM1" title="GLIB_009:NM.1">removeLoops</a> of <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">H</font> is    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a> </span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">G9</font> =  the    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">G8</font> = <font color="Maroon" title="c1">G</font> as    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T35" target="_self" onmouseover="rs('glenum00/T35')" onmouseout="rh()">Th35</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">H9</font> =  the    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="234_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"> (  the    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a>  is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c4">G8</font> &amp;  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c2">H</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>  the    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a>  &amp;  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c2">H</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>  the    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a>  )</span></a><div class="add"><span class="kw">thus </span><a NAME="E1:234_1"/>
 the    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a>  is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c4">G8</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glibpre1.html#T119" onmouseover="rs('glibpre1/T119')" onmouseout="rh()">GLIBPRE1:119</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"> (  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c2">H</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>  the    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a>  &amp;  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c2">H</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>  the    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a>  )</span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="234_1_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="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c2">H</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a>  the    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c2">H</font> implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a>  the    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a>  )</span><br/><span class="kw">set </span><font color="Maroon" title="c7">H8</font> =  the    <a href="glenum00.html#M1" title="GLENUM00:mode.1">Element</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a> ;<br/><span class="kw">assume </span><a NAME="E1:234_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c2">H</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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a>  the    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a> </span><br/><a NAME="E2:234_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E142">A1</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="E3:234_1_1"/>
 the    <a href="glenum00.html#M1" title="GLENUM00:mode.1">Element</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a>  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="glenum00.html#T168" target="_self" onmouseover="rs('glenum00/T168')" onmouseout="rh()">Th168</a></span>;<br/><a NAME="E4:234_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E143">A2</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the    <a href="glenum00.html#M1" title="GLENUM00:mode.1">Element</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:234_1_1"><span class="lab"><font color="Green" title="E142">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>;<br/><a NAME="E5:234_1_1"/>
 the    <a href="glenum00.html#M1" title="GLENUM00:mode.1">Element</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a>  is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of  the    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_014.html#T21" onmouseover="rs('glib_014/T21')" onmouseout="rh()">GLIB_014:21</a></span>;<br/><a NAME="E6:234_1_1"/><span class="kw">then </span>
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the    <a href="glenum00.html#M1" title="GLENUM00:mode.1">Element</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a>  <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>  the    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D32" onmouseover="rs('glib_000/D32')" onmouseout="rh()">GLIB_000:def 32</a></span>;<br/><span class="kw">hence </span><a NAME="E7:234_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a>  the    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:234_1_1"><span class="lab"><font color="Green" title="E143">A2</font></span></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="E3:234_1"/>
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c2">H</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>  the    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a> 
 <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>; <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_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c2">H</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>  the    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a> </span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="234_1_2"><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="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c2">H</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a>  the    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c2">H</font> implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a>  the    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:234_1_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c2">H</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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a>  the    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a> </span><br/><a NAME="E2:234_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T53" onmouseover="rs('glib_000/T53')" onmouseout="rh()">GLIB_000:53</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">T</font> being   <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>   <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</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><span class="kw"> such that </span><br/><a NAME="E4:234_1_2"/><span class="lab"><font color="Green" title="E142">A3</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c7">T</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T167" target="_self" onmouseover="rs('glenum00/T167')" onmouseout="rh()">Th167</a></span>;<br/><a NAME="E5:234_1_2"/>
<font color="Maroon" title="c7">T</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T168" target="_self" onmouseover="rs('glenum00/T168')" onmouseout="rh()">Th168</a></span>;<br/><a NAME="E6:234_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">T</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of  the    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_014.html#T21" onmouseover="rs('glib_014/T21')" onmouseout="rh()">GLIB_014:21</a></span>;<br/><a NAME="E7:234_1_2"/><span class="kw">then </span>
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c7">T</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>  the    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D32" onmouseover="rs('glib_000/D32')" onmouseout="rh()">GLIB_000:def 32</a></span>;<br/><span class="kw">hence </span><a NAME="E8:234_1_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a>  the    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:234_1_2"><span class="lab"><font color="Green" title="E142">A3</font></span></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="E5:234_1"/>
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c2">H</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>  the    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a> 
 <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>; <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="E3:234"/><span class="kw">then </span><span class="lab"><font color="Green" title="E142">A4</font></span>: 
<font color="Maroon" title="c2">H</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of  the    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T44" onmouseover="rs('glib_000/T44')" onmouseout="rh()">GLIB_000:44</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="234_2"><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">H2</font> being    <a href="glenum00.html#M1" title="GLENUM00:mode.1">Element</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a>   ex <font color="Olive" title="b2">H1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> st <font color="Olive" title="b1">H2</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Olive" title="b2">H1</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">H2</font> be    <a href="glenum00.html#M1" title="GLENUM00:mode.1">Element</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</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">  ex <font color="Olive" title="b1">H1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> st <font color="Maroon" title="c6">H2</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Olive" title="b1">H1</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c7">H1</font> = <font color="Maroon" title="c2">H</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">H</font></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>;<br/><span class="kw">take </span><font color="Maroon" title="c8">H1</font> = <font color="Maroon" title="c7">H1</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="c6">H2</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c8">H1</font></span><br/><a NAME="E2:234_2"/>
<font color="Maroon" title="c2">H</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T176" target="_self" onmouseover="rs('glenum00/T176')" onmouseout="rh()">Th176</a></span>;<br/><span class="kw">hence </span><a NAME="E3:234_2"/>
<font color="Maroon" title="c6">H2</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c8">H1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T168" target="_self" onmouseover="rs('glenum00/T168')" onmouseout="rh()">Th168</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="E5:234"/><span class="kw">then </span>
 the    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a>  is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of  the    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glibpre1.html#T118" onmouseover="rs('glibpre1/T118')" onmouseout="rh()">GLIBPRE1:118</a></span>;<br/>
<a NAME="E6:234"/><span class="kw">then </span>
 the    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a>  is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c2">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T91" onmouseover="rs('glib_000/T91')" onmouseout="rh()">GLIB_000:91</a>, <a class="ref" href="glib_014.html#T24" onmouseover="rs('glib_014/T24')" onmouseout="rh()">GLIB_014:24</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:234"/>
<font color="Maroon" title="c2">H</font> is    <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:234"><span class="lab"><font color="Green" title="E142">A4</font></span></a>, <a class="ref" href="glib_000.html#T87" onmouseover="rs('glib_000/T87')" onmouseout="rh()">GLIB_000:87</a>, <a class="ref" href="glib_014.html#T22" onmouseover="rs('glib_014/T22')" onmouseout="rh()">GLIB_014:22</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>
