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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"> ( <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>  iff  <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#K14" title="GLENUM00:func.14">.allSpanningTrees()</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><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"> (  <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#K14" title="GLENUM00:func.14">.allSpanningTrees()</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> implies <font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:237_1"/><span class="lab"><font color="Green" title="E142">A1</font></span>: 
<font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</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#K14" title="GLENUM00:func.14">.allSpanningTrees()</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/><a NAME="E2:237_1"/><span class="lab"><font color="Green" title="E143">A2</font></span>: 
 <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#K10" title="GLENUM00:func.10">.allSpanningForests()</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="glenum00.html#T123" target="_self" onmouseover="rs('glenum00/T123')" onmouseout="rh()">Th123</a></span>;<br/><a NAME="E3:237_1"/>
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K11" title="GLENUM00:func.11">.allConnectedSG()</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#K10" title="GLENUM00:func.10">.allSpanningForests()</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T170" target="_self" onmouseover="rs('glenum00/T170')" onmouseout="rh()">Th170</a></span>;<br/><a NAME="E4:237_1"/><span class="kw">then </span>
<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> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K10" title="GLENUM00:func.10">.allSpanningForests()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" onmouseover="rs('xboole_1/T17')" onmouseout="rh()">XBOOLE_1:17</a></span>;<br/><a NAME="E5:237_1"/><span class="kw">then </span>
 <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#K14" title="GLENUM00:func.14">.allSpanningTrees()</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="txt" href="#E2:237_1"><span class="lab"><font color="Green" title="E143">A2</font></span></a>, <a class="ref" href="glibpre1.html#T115" onmouseover="rs('glibpre1/T115')" onmouseout="rh()">GLIBPRE1:115</a></span>;<br/><span class="kw">hence </span><a NAME="E6:237_1"/>
 <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#K14" title="GLENUM00:func.14">.allSpanningTrees()</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="txt" href="#E1:237_1"><span class="lab"><font color="Green" title="E142">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" onmouseover="rs('zfmisc_1/T33')" onmouseout="rh()">ZFMISC_1:33</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="E2:237"/>
 <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#K14" title="GLENUM00:func.14">.allSpanningTrees()</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>
 ; <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="c1">G</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> </span><br/>

<span class="kw">hence </span><a NAME="E3:237"/>
<font color="Maroon" title="c1">G</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</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"> verum</span><br/>


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