<?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#K12" title="GLENUM00:func.12">.allTrees()</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#K12" title="GLENUM00:func.12">.allTrees()</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">set </span><font color="Maroon" title="c2">H</font> =  the   <a href="glib_009.html#NM1" title="GLIB_009:NM.1">removeLoops</a> of <font color="Maroon" title="c1">G</font>;<br/><span class="kw">assume </span><a NAME="E1:200_1"/><span class="lab"><font color="Green" title="E120">A1</font></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="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#K12" title="GLENUM00:func.12">.allTrees()</a>  is  <a href="glib_014.html#V12" title="GLIB_014:attr.12">edgeless</a> </span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="200_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">H</font> being   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>  st <font color="Olive" title="b1">H</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a>  holds <br/><font color="Olive" title="b1">H</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">H</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="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#K12" title="GLENUM00:func.12">.allTrees()</a>  implies <font color="Maroon" title="c3">H</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:200_1_1"/>
<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#K12" title="GLENUM00:func.12">.allTrees()</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="c3">H</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a> </span><br/><a NAME="E2:200_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E121">A2</font></span>: 
<font color="Maroon" title="c3">H</font> is   <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</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="ref" href="glenum00.html#T138" target="_self" onmouseover="rs('glenum00/T138')" onmouseout="rh()">Th138</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="200_1_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">  not  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c3">H</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:200_1_1_1"/>
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c3">H</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:200_1_1_1"/><span class="lab"><font color="Green" title="E122">A3</font></span>: 
<font color="Maroon" title="c4">e</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="c3">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</a></span>;<br/><a NAME="E4:200_1_1_1"/>
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c3">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> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:200_1_1"><span class="lab"><font color="Green" title="E121">A2</font></span></a>, <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">then </span><span class="kw">consider </span><font color="Maroon" title="c5">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:200_1_1_1"/><span class="lab"><font color="Green" title="E123">A4</font></span>: 
<font color="Maroon" title="c4">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:200_1"><span class="lab"><font color="Green" title="E120">A1</font></span></a>, <a class="txt" href="#E3:200_1_1_1"><span class="lab"><font color="Green" title="E122">A3</font></span></a>, <a class="ref" href="glib_009.html#D2" onmouseover="rs('glib_009/D2')" onmouseout="rh()">GLIB_009:def 2</a></span>;<br/><a NAME="E7:200_1_1_1"/>
( <font color="Maroon" title="c4">e</font> is    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  &amp; <font color="Maroon" title="c5">v</font> is    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" onmouseover="rs('tarski/T1')" onmouseout="rh()">TARSKI:1</a></span>;<br/><a NAME="E8:200_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c3">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:200_1_1"><span class="lab"><font color="Green" title="E121">A2</font></span></a>, <a class="txt" href="#E3:200_1_1_1"><span class="lab"><font color="Green" title="E122">A3</font></span></a>, <a class="txt" href="#E6:200_1_1_1"><span class="lab"><font color="Green" title="E123">A4</font></span></a>, <a class="ref" href="glib_000.html#T73" onmouseover="rs('glib_000/T73')" onmouseout="rh()">GLIB_000:73</a></span>;<br/><span class="kw">hence </span><a NAME="E9:200_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:200_1_1"><span class="lab"><font color="Green" title="E121">A2</font></span></a>, <a class="ref" href="glib_000.html#T18" onmouseover="rs('glib_000/T18')" onmouseout="rh()">GLIB_000:18</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="E4:200_1_1"/>
<font color="Maroon" title="c3">H</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">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"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:200_1"/>
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</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="glib_014.html#D12" onmouseover="rs('glib_014/D12')" onmouseout="rh()">GLIB_014:def 12</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="E2:200"/><span class="lab"><font color="Green" title="E120">A5</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</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="200_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="E1:200_2"/><span class="lab"><font color="Green" title="E121">A6</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="#E1:200_2"><span class="lab"><font color="Green" title="E121">A6</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c5">H9</font> =  <a href="glenum00.html#K4" title="GLENUM00:func.4">createGraph</a> <font color="Maroon" title="c4">e</font>;<br/><span class="kw">assume </span><a NAME="E4:200_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="E5:200_2"/><span class="kw">then </span>
 <a href="glenum00.html#K4" title="GLENUM00:func.4">createGraph</a> <font color="Maroon" title="c4">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T143" target="_self" onmouseover="rs('glenum00/T143')" onmouseout="rh()">Th143</a></span>;<br/><span class="kw">hence </span><a NAME="E6:200_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:200"><span class="lab"><font color="Green" title="E120">A5</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="E4:200"/><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="E5:200"/>
 <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>
