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<a NAME="E1:212_1_1"/><span class="lab"><font color="Green" title="E127">A1</font></span>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="glenum00.html#K13" title="GLENUM00:func.13">SubtreeRel</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#T160" target="_self" onmouseover="rs('glenum00/T160')" onmouseout="rh()">Th160</a></span>;<br/>
<a NAME="E2:212_1_1"/>
 <a href="glenum00.html#K13" title="GLENUM00:func.13">SubtreeRel</a> <font color="Maroon" title="c1">G</font> <a href="orders_1.html#R2" title="ORDERS_1:pred.2">partially_orders</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#T161" target="_self" onmouseover="rs('glenum00/T161')" onmouseout="rh()">Th161</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E4:212_1_1"/><span class="lab"><font color="Green" title="E128">A2</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="orders_1.html#R6" title="ORDERS_1:pred.6">is_maximal_in</a>  <a href="glenum00.html#K13" title="GLENUM00:func.13">SubtreeRel</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:212_1_1"><span class="lab"><font color="Green" title="E127">A1</font></span></a>, <a class="ref" href="glenum00.html#T166" target="_self" onmouseover="rs('glenum00/T166')" onmouseout="rh()">Th166</a>, <a class="ref" href="orders_1.html#T63" onmouseover="rs('orders_1/T63')" onmouseout="rh()">ORDERS_1:63</a></span>;<br/>
<a NAME="E5:212_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="glenum00.html#K13" title="GLENUM00:func.13">SubtreeRel</a> <font color="Maroon" title="c1">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:212_1_1"><span class="lab"><font color="Green" title="E128">A2</font></span></a>, <a class="ref" href="orders_1.html#D12" onmouseover="rs('orders_1/D12')" onmouseout="rh()">ORDERS_1:def 12</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">T</font> = <font color="Maroon" title="c2">x</font> as   <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">by</span> <span class="lab"><a class="txt" href="#E1:212_1_1"><span class="lab"><font color="Green" title="E127">A1</font></span></a>, <a class="ref" href="glenum00.html#T138" target="_self" onmouseover="rs('glenum00/T138')" onmouseout="rh()">Th138</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">T</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="c3">T</font> is  <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  &amp; <font color="Maroon" title="c3">T</font> is  <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>  &amp; <font color="Maroon" title="c3">T</font> is  <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>  )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="212_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:212_1_1_1"/>
( <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  or  not <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="212_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:212_1_1_1_1"/>
<font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</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">T</font> is  <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  &amp; <font color="Maroon" title="c3">T</font> is  <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>  &amp; <font color="Maroon" title="c3">T</font> is  <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>  )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:212_1_1_1_1"/>
( <font color="Maroon" title="c3">T</font> is  <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  &amp; <font color="Maroon" title="c3">T</font> is  <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>  &amp; <font color="Maroon" title="c3">T</font> is  <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T121" onmouseover="rs('glib_000/T121')" onmouseout="rh()">GLIB_000:121</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="212_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:212_1_1_1_2"/><span class="lab"><font color="Green" title="E129">A3</font></span>: 
 not <font color="Maroon" title="c1">G</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</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">T</font> is  <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  &amp; <font color="Maroon" title="c3">T</font> is  <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>  &amp; <font color="Maroon" title="c3">T</font> is  <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>  )</span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="212_1_1_1_2_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"> <font color="Maroon" title="c3">T</font> is  <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:212_1_1_1_2_1"/>
 not <font color="Maroon" title="c3">T</font> is  <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</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">v</font>, <font color="Maroon" title="c5">e</font>, <font color="Maroon" title="c6">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:212_1_1_1_2_1"/><span class="lab"><font color="Green" title="E130">A4</font></span>: 
( <font color="Maroon" title="c4">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c5">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c1">G</font> &amp;  not <font color="Maroon" title="c5">e</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="c3">T</font> )
 <span class="kw">and </span><br/><a NAME="E4:212_1_1_1_2_1"/><span class="lab"><font color="Green" title="E131">A5</font></span>: 
 for <font color="Olive" title="b1">G3</font> being    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c6">w</font> holds  <font color="Olive" title="b1">G3</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</font>
 <span class="kw">and </span><br/><a NAME="E5:212_1_1_1_2_1"/><span class="lab"><font color="Green" title="E132">A6</font></span>: 
( ( <font color="Maroon" title="c4">v</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="c3">T</font> &amp;  not <font color="Maroon" title="c6">w</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="c3">T</font> ) or (  not <font color="Maroon" title="c4">v</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="c3">T</font> &amp; <font color="Maroon" title="c6">w</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="c3">T</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:212_1_1_1_2"><span class="lab"><font color="Green" title="E129">A3</font></span></a>, <a class="ref" href="glibpre1.html#T61" onmouseover="rs('glibpre1/T61')" onmouseout="rh()">GLIBPRE1:61</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c7">G3</font> =  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c6">w</font>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">G3</font> =  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c6">w</font> as   <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">by</span> <span class="lab"><a class="txt" href="#E4:212_1_1_1_2_1"><span class="lab"><font color="Green" title="E131">A5</font></span></a></span>;<br/><a NAME="E7:212_1_1_1_2_1"/>
<font color="Maroon" title="c5">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c8">G3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:212_1_1_1_2_1"><span class="lab"><font color="Green" title="E130">A4</font></span></a>, <a class="txt" href="#E5:212_1_1_1_2_1"><span class="lab"><font color="Green" title="E132">A6</font></span></a>, <a class="ref" href="glib_006.html#T131" onmouseover="rs('glib_006/T131')" onmouseout="rh()">GLIB_006:131</a>, <a class="ref" href="glib_006.html#T132" onmouseover="rs('glib_006/T132')" onmouseout="rh()">GLIB_006:132</a></span>;<br/><a NAME="E8:212_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E133">A7</font></span>: 
<font color="Maroon" title="c3">T</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">G3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:212_1_1_1_2_1"><span class="lab"><font color="Green" title="E130">A4</font></span></a>, <a class="ref" href="glib_000.html#D14" onmouseover="rs('glib_000/D14')" onmouseout="rh()">GLIB_000:def 14</a></span>;<br/><a NAME="E9:212_1_1_1_2_1"/><span class="lab"><font color="Green" title="E134">A8</font></span>: 
<font color="Maroon" title="c8">G3</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><a href="glenum00.html#K13" title="GLENUM00:func.13">SubtreeRel</a> <font color="Maroon" title="c1">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:212_1_1"><span class="lab"><font color="Green" title="E127">A1</font></span></a>, <a class="ref" href="glenum00.html#T138" target="_self" onmouseover="rs('glenum00/T138')" onmouseout="rh()">Th138</a></span>;<br/><a NAME="E10:212_1_1_1_2_1"/>
<font color="Maroon" title="c3">T</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c8">G3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_006.html#T57" onmouseover="rs('glib_006/T57')" onmouseout="rh()">GLIB_006:57</a></span>;<br/><a NAME="E11:212_1_1_1_2_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c8">G3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glenum00.html#K13" title="GLENUM00:func.13">SubtreeRel</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T159" target="_self" onmouseover="rs('glenum00/T159')" onmouseout="rh()">Th159</a></span>;<br/><span class="kw">hence </span><a NAME="E12:212_1_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:212_1_1"><span class="lab"><font color="Green" title="E128">A2</font></span></a>, <a class="txt" href="#E8:212_1_1_1_2_1"><span class="lab"><font color="Green" title="E133">A7</font></span></a>, <a class="txt" href="#E9:212_1_1_1_2_1"><span class="lab"><font color="Green" title="E134">A8</font></span></a>, <a class="ref" href="orders_1.html#D12" onmouseover="rs('orders_1/D12')" onmouseout="rh()">ORDERS_1: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">hence </span><a NAME="E3:212_1_1_1_2"/>
( <font color="Maroon" title="c3">T</font> is  <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  &amp; <font color="Maroon" title="c3">T</font> is  <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>  &amp; <font color="Maroon" title="c3">T</font> is  <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</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></div><span class="kw">end;</span></div>

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