<?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"> (  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a>  iff <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K10" title="GLENUM00:func.10">.allSpanningForests()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  )</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="c1">G</font> <a href="glenum00.html#K10" title="GLENUM00:func.10">.allSpanningForests()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a>  implies  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:163_1"/>
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</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> <a href="glenum00.html#K10" title="GLENUM00:func.10">.allSpanningForests()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a> </span><br/><a NAME="E2:163_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A1</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K9" title="GLENUM00:func.9">.allForests()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T99" target="_self" onmouseover="rs('glenum00/T99')" onmouseout="rh()">Th99</a></span>;<br/><a NAME="E3:163_1"/>
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K10" title="GLENUM00:func.10">.allSpanningForests()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K9" title="GLENUM00:func.9">.allForests()</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T103" target="_self" onmouseover="rs('glenum00/T103')" onmouseout="rh()">Th103</a></span>;<br/><span class="kw">hence </span><a NAME="E4:163_1"/>
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K10" title="GLENUM00:func.10">.allSpanningForests()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:163_1"><span class="lab"><font color="Green" title="E94">A1</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">assume </span><a NAME="E2:163"/><span class="lab"><font color="Green" title="E94">A2</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K10" title="GLENUM00:func.10">.allSpanningForests()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</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_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a> </span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="163_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#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a> </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_000.html#K7" title="GLIB_000:func.7">the_Edges_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> <font color="Maroon" title="c1">G</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:163_2"/><span class="lab"><font color="Green" title="E95">A3</font></span>: 
<font color="Maroon" title="c2">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> <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> <font color="Maroon" title="c1">G</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a> </span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">G9</font> = <font color="Maroon" title="c1">G</font> as   non  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c4">e</font> = <font color="Maroon" title="c2">x</font> as   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c3">G9</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:163_2"><span class="lab"><font color="Green" title="E95">A3</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c5">H9</font> =  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_006.html#M2" title="GLIB_006:mode.2">addVertices</a> of  <a href="glenum00.html#K4" title="GLENUM00:func.4">createGraph</a> <font color="Maroon" title="c4">e</font>, <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>;<br/><span class="kw">assume </span><a NAME="E4:163_2"/>
 not <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="glib_009.html#K1" title="GLIB_009:func.1">.loops()</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"> contradiction</span><br/><a NAME="E5:163_2"/><span class="kw">then </span>
 the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_006.html#M2" title="GLIB_006:mode.2">addVertices</a> of  <a href="glenum00.html#K4" title="GLENUM00:func.4">createGraph</a> <font color="Maroon" title="c4">e</font>, <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#R2" title="TARSKI:pred.2">in</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="glenum00.html#T107" target="_self" onmouseover="rs('glenum00/T107')" onmouseout="rh()">Th107</a></span>;<br/><span class="kw">hence </span><a NAME="E6:163_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:163"><span class="lab"><font color="Green" title="E94">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>
<a NAME="E4:163"/><span class="kw">then </span>
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</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>;<br/>
<span class="kw">hence </span><a NAME="E5:163"/>
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a> 
 <span class="kw">by</span> <span class="lab"><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/>


</div>
