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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> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a>  <a href="orders_1.html#R4" title="ORDERS_1:pred.4">has_upper_Zorn_property_wrt</a>  <a href="glenum00.html#K13" title="GLENUM00:func.13">SubtreeRel</a> <font color="Maroon" title="c1">G</font></span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="211_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">Y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a>  &amp; <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="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Olive" title="b1">Y</font> is  <a href="orders_1.html#V3" title="ORDERS_1:attr.3">being_linear-order</a>  holds <br/> ex <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b2">x</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>  &amp; (  for <font color="Olive" title="b3">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b3">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">Y</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">y</font>,<font color="Olive" title="b2">x</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></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">Y</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a>  &amp; <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="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">Y</font> is  <a href="orders_1.html#V3" title="ORDERS_1:attr.3">being_linear-order</a>  implies  ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></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>  &amp; (  for <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive">b<sub>3</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">x</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive" title="b2">y</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><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:211_1"/><span class="lab"><font color="Green" title="E127">A1</font></span>: 
<font color="Maroon" title="c2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a> 
 <span class="kw">and </span><br/><a NAME="E2:211_1"/><span class="lab"><font color="Green" title="E128">A2</font></span>: 
<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="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">Y</font> is  <a href="orders_1.html#V3" title="ORDERS_1:attr.3">being_linear-order</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">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></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>  &amp; (  for <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive">b<sub>3</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">x</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive" title="b2">y</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><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="211_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="E1:211_1_1"/>
( <font color="Maroon" title="c2">Y</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>  or <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="211_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:211_1_1_1"/><span class="lab"><font color="Green" title="E129">A3</font></span>: 
<font color="Maroon" title="c2">Y</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">  ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></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>  &amp; (  for <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive">b<sub>3</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">x</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive" title="b2">y</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><br/><div class="add"><a NAME="E2:211_1_1_1"/><span class="lab"><font color="Green" title="E130">A4</font></span>: 
<font color="Maroon" title="c2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <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="txt" href="#E1:211_1"><span class="lab"><font color="Green" title="E127">A1</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">S</font> = <font color="Maroon" title="c2">Y</font> as   <a href="glib_014.html#NM1" title="GLIB_014:NM.1">GraphUnionSet</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:211_1_1_1"><span class="lab"><font color="Green" title="E129">A3</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c4">T</font> =  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c3">S</font>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="211_1_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">  for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c3">S</font>  ex <font color="Olive" title="b3">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c3">S</font> st <font color="Olive" title="b3">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">v2</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">v1</font>, <font color="Maroon" title="c6">v2</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c3">S</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">  ex <font color="Olive" title="b1">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c3">S</font> st <font color="Olive">b<sub>3</sub></font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Olive" title="b1">W</font>,<font color="Olive">b<sub>2</sub></font></span><br/><a NAME="E1:211_1_1_1_1"/><span class="lab"><font color="Green" title="E131">A5</font></span>: 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a>  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="glib_014.html#K1" title="GLIB_014:func.1">the_Vertices_of</a> <font color="Maroon" title="c3">S</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_014.html#D25" onmouseover="rs('glib_014/D25')" onmouseout="rh()">GLIB_014:def 25</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">V1</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:211_1_1_1_1"/><span class="lab"><font color="Green" title="E132">A6</font></span>: 
( <font color="Maroon" title="c5">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">V1</font> &amp; <font color="Maroon" title="c7">V1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_014.html#K1" title="GLIB_014:func.1">the_Vertices_of</a> <font color="Maroon" title="c3">S</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c8">H1</font> being   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a><span class="kw"> such that </span><br/><a NAME="E5:211_1_1_1_1"/><span class="lab"><font color="Green" title="E133">A7</font></span>: 
( <font color="Maroon" title="c8">H1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">S</font> &amp; <font color="Maroon" title="c7">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c8">H1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:211_1_1_1_1"><span class="lab"><font color="Green" title="E132">A6</font></span></a>, <a class="ref" href="glib_014.html#D14" onmouseover="rs('glib_014/D14')" onmouseout="rh()">GLIB_014:def 14</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">V2</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E7:211_1_1_1_1"/><span class="lab"><font color="Green" title="E134">A8</font></span>: 
( <font color="Maroon" title="c6">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">V2</font> &amp; <font color="Maroon" title="c9">V2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_014.html#K1" title="GLIB_014:func.1">the_Vertices_of</a> <font color="Maroon" title="c3">S</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:211_1_1_1_1"><span class="lab"><font color="Green" title="E131">A5</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">H2</font> being   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a><span class="kw"> such that </span><br/><a NAME="E9:211_1_1_1_1"/><span class="lab"><font color="Green" title="E135">A9</font></span>: 
( <font color="Maroon" title="c10">H2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">S</font> &amp; <font color="Maroon" title="c9">V2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c10">H2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:211_1_1_1_1"><span class="lab"><font color="Green" title="E134">A8</font></span></a>, <a class="ref" href="glib_014.html#D14" onmouseover="rs('glib_014/D14')" onmouseout="rh()">GLIB_014:def 14</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">H1</font> = <font color="Maroon" title="c8">H1</font>, <font color="Maroon" title="c12">H2</font> = <font color="Maroon" title="c10">H2</font> as    <a href="glenum00.html#M1" title="GLENUM00:mode.1">Element</a> of <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="txt" href="#E1:211_1"><span class="lab"><font color="Green" title="E127">A1</font></span></a>, <a class="txt" href="#E5:211_1_1_1_1"><span class="lab"><font color="Green" title="E133">A7</font></span></a>, <a class="txt" href="#E9:211_1_1_1_1"><span class="lab"><font color="Green" title="E135">A9</font></span></a></span>;<br/><a NAME="E11:211_1_1_1_1"/>
<font color="Maroon" title="c2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="#E1:211_1"><span class="lab"><font color="Green" title="E127">A1</font></span></a>, <a class="ref" href="glenum00.html#T160" target="_self" onmouseover="rs('glenum00/T160')" onmouseout="rh()">Th160</a></span>;<br/><a NAME="E12:211_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E136">A10</font></span>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">field</a> <span class="p1">(<span class="default"><span class="p2">(<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="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_1.html#T71" onmouseover="rs('orders_1/T71')" onmouseout="rh()">ORDERS_1:71</a></span>;<br/><a NAME="E13:211_1_1_1_1"/>
( <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="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">Y</font> is  <a href="relat_2.html#V1" title="RELAT_2:attr.1">reflexive</a>  &amp; <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="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">Y</font> is  <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:211_1"><span class="lab"><font color="Green" title="E128">A2</font></span></a>, <a class="ref" href="orders_1.html#D6" onmouseover="rs('orders_1/D6')" onmouseout="rh()">ORDERS_1:def 6</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="211_1_1_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="E1:211_1_1_1_1_1"/><span class="kw">then </span>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">H1</font>,<font color="Maroon" title="c12">H2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">Y</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">H2</font>,<font color="Maroon" title="c11">H1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">Y</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:211_1_1_1_1"><span class="lab"><font color="Green" title="E133">A7</font></span></a>, <a class="txt" href="#E9:211_1_1_1_1"><span class="lab"><font color="Green" title="E135">A9</font></span></a>, <a class="txt" href="#E12:211_1_1_1_1"><span class="lab"><font color="Green" title="E136">A10</font></span></a>, <a class="ref" href="relat_2.html#D7" onmouseover="rs('relat_2/D7')" onmouseout="rh()">RELAT_2:def 7</a>, <a class="ref" href="relat_2.html#D15" onmouseover="rs('relat_2/D15')" onmouseout="rh()">RELAT_2:def 15</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="211_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:211_1_1_1_1_1_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">H1</font>,<font color="Maroon" title="c12">H2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">Y</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">  ex <font color="Olive" title="b1">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c3">S</font> st <font color="Olive">b<sub>3</sub></font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Olive" title="b1">W</font>,<font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><a NAME="E2:211_1_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">H1</font>,<font color="Maroon" title="c12">H2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI: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="mmlquer2.html#T4" onmouseover="rs('mmlquer2/T4')" onmouseout="rh()">MMLQUER2:4</a></span>;<br/><a NAME="E3:211_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">H1</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c12">H2</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/><a NAME="E4:211_1_1_1_1_1_1"/><span class="kw">then </span>
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c11">H1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">H2</font>
 <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">then </span><span class="kw">consider </span><font color="Maroon" title="c13">W2</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c12">H2</font><span class="kw"> such that </span><br/><a NAME="E6:211_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E137">A11</font></span>: 
<font color="Maroon" title="c13">W2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:211_1_1_1_1"><span class="lab"><font color="Green" title="E132">A6</font></span></a>, <a class="txt" href="#E5:211_1_1_1_1"><span class="lab"><font color="Green" title="E133">A7</font></span></a>, <a class="txt" href="#E7:211_1_1_1_1"><span class="lab"><font color="Green" title="E134">A8</font></span></a>, <a class="txt" href="#E9:211_1_1_1_1"><span class="lab"><font color="Green" title="E135">A9</font></span></a>, <a class="ref" href="glib_002.html#D1" onmouseover="rs('glib_002/D1')" onmouseout="rh()">GLIB_002:def 1</a></span>;<br/><a NAME="E7:211_1_1_1_1_1_1"/>
<font color="Maroon" title="c12">H2</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c3">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:211_1_1_1_1"><span class="lab"><font color="Green" title="E135">A9</font></span></a>, <a class="ref" href="glib_014.html#T21" onmouseover="rs('glib_014/T21')" onmouseout="rh()">GLIB_014:21</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">W</font> = <font color="Maroon" title="c13">W2</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c3">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T167" onmouseover="rs('glib_001/T167')" onmouseout="rh()">GLIB_001:167</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c15">W</font> = <font color="Maroon" title="c14">W</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="c15">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font></span><br/><span class="kw">thus </span><a NAME="E9:211_1_1_1_1_1_1"/>
<font color="Maroon" title="c15">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:211_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E137">A11</font></span></a>, <a class="ref" href="glib_001.html#T19" onmouseover="rs('glib_001/T19')" onmouseout="rh()">GLIB_001:19</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="211_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:211_1_1_1_1_1_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">H2</font>,<font color="Maroon" title="c11">H1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">Y</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">  ex <font color="Olive" title="b1">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c3">S</font> st <font color="Olive">b<sub>3</sub></font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Olive" title="b1">W</font>,<font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><a NAME="E2:211_1_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">H2</font>,<font color="Maroon" title="c11">H1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="tarski.html#R2" title="TARSKI: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="mmlquer2.html#T4" onmouseover="rs('mmlquer2/T4')" onmouseout="rh()">MMLQUER2:4</a></span>;<br/><a NAME="E3:211_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">H2</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c11">H1</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/><a NAME="E4:211_1_1_1_1_1_2"/><span class="kw">then </span>
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">H2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c11">H1</font>
 <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">then </span><span class="kw">consider </span><font color="Maroon" title="c13">W1</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c11">H1</font><span class="kw"> such that </span><br/><a NAME="E6:211_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E137">A12</font></span>: 
<font color="Maroon" title="c13">W1</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:211_1_1_1_1"><span class="lab"><font color="Green" title="E132">A6</font></span></a>, <a class="txt" href="#E5:211_1_1_1_1"><span class="lab"><font color="Green" title="E133">A7</font></span></a>, <a class="txt" href="#E7:211_1_1_1_1"><span class="lab"><font color="Green" title="E134">A8</font></span></a>, <a class="txt" href="#E9:211_1_1_1_1"><span class="lab"><font color="Green" title="E135">A9</font></span></a>, <a class="ref" href="glib_002.html#D1" onmouseover="rs('glib_002/D1')" onmouseout="rh()">GLIB_002:def 1</a></span>;<br/><a NAME="E7:211_1_1_1_1_1_2"/>
<font color="Maroon" title="c11">H1</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c3">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:211_1_1_1_1"><span class="lab"><font color="Green" title="E133">A7</font></span></a>, <a class="ref" href="glib_014.html#T21" onmouseover="rs('glib_014/T21')" onmouseout="rh()">GLIB_014:21</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">W</font> = <font color="Maroon" title="c13">W1</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c3">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T167" onmouseover="rs('glib_001/T167')" onmouseout="rh()">GLIB_001:167</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c15">W</font> = <font color="Maroon" title="c14">W</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="c15">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font></span><br/><span class="kw">thus </span><a NAME="E9:211_1_1_1_1_1_2"/>
<font color="Maroon" title="c15">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c6">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:211_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E137">A12</font></span></a>, <a class="ref" href="glib_001.html#T19" onmouseover="rs('glib_001/T19')" onmouseout="rh()">GLIB_001:19</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">T</font> =  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_014.html#M1" title="GLIB_014:mode.1">GraphUnion</a> of <font color="Maroon" title="c3">S</font> as   <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> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D1" onmouseover="rs('glib_002/D1')" onmouseout="rh()">GLIB_002:def 1</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="211_1_1_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">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c5">T</font>  holds  not <font color="Olive" title="b1">W</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c6">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c5">T</font><span class="kw"> such that </span><a NAME="E2:211_1_1_1_2"/><span class="lab"><font color="Green" title="E131">A13</font></span>: 
<font color="Maroon" title="c6">W</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-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"> contradiction</span><br/><a NAME="E3:211_1_1_1_2"/>
<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="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glenum00.html#K6" title="GLENUM00:func.6">SubgraphRel</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c3">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:211_1"><span class="lab"><font color="Green" title="E127">A1</font></span></a>, <a class="ref" href="wellord1.html#T22" onmouseover="rs('wellord1/T22')" onmouseout="rh()">WELLORD1:22</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">H</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">S</font><span class="kw"> such that </span><br/><a NAME="E5:211_1_1_1_2"/><span class="lab"><font color="Green" title="E132">A14</font></span>: 
<font color="Maroon" title="c6">W</font> is    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c7">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:211_1"><span class="lab"><font color="Green" title="E128">A2</font></span></a>, <a class="txt" href="#E2:211_1_1_1"><span class="lab"><font color="Green" title="E130">A4</font></span></a>, <a class="ref" href="glenum00.html#T44" target="_self" onmouseover="rs('glenum00/T44')" onmouseout="rh()">Th44</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">W</font> = <font color="Maroon" title="c6">W</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c7">H</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:211_1_1_1_2"><span class="lab"><font color="Green" title="E132">A14</font></span></a></span>;<br/><a NAME="E7:211_1_1_1_2"/>
<font color="Maroon" title="c8">W</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:211_1_1_1_2"><span class="lab"><font color="Green" title="E131">A13</font></span></a>, <a class="ref" href="glib_006.html#T24" onmouseover="rs('glib_006/T24')" onmouseout="rh()">GLIB_006:24</a></span>;<br/><span class="kw">hence </span><a NAME="E8:211_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:211_1"><span class="lab"><font color="Green" title="E127">A1</font></span></a>, <a class="ref" href="glib_002.html#D2" onmouseover="rs('glib_002/D2')" onmouseout="rh()">GLIB_002:def 2</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="E7:211_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">T</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D2" onmouseover="rs('glib_002/D2')" onmouseout="rh()">GLIB_002:def 2</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">T</font> = <font color="Maroon" title="c5">T</font> as   <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a> ;<br/><a NAME="E9:211_1_1_1"/><span class="lab"><font color="Green" title="E131">A15</font></span>: 
<font color="Maroon" title="c3">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <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="txt" href="#E1:211_1"><span class="lab"><font color="Green" title="E127">A1</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/><a NAME="E10:211_1_1_1"/>
<font color="Maroon" title="c1">G</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#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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">T</font> = <font color="Maroon" title="c6">T</font> as   <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="txt" href="#E9:211_1_1_1"><span class="lab"><font color="Green" title="E131">A15</font></span></a>, <a class="ref" href="glibpre1.html#T119" onmouseover="rs('glibpre1/T119')" onmouseout="rh()">GLIBPRE1:119</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">T</font> as    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  ;<br/><span class="kw">take </span><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c8">x</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="c9">x</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>  &amp; (  for <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Y</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c9">x</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><br/><span class="kw">thus </span><a NAME="E13:211_1_1_1"/>
<font color="Maroon" title="c9">x</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#T138" target="_self" onmouseover="rs('glenum00/T138')" onmouseout="rh()">Th138</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">  for <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Y</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c9">x</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><br/><span class="kw">let </span><font color="Maroon" title="c10">y</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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="c10">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Y</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c9">x</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><br/><span class="kw">assume </span><a NAME="E14:211_1_1_1"/><span class="lab"><font color="Green" title="E132">A16</font></span>: 
<font color="Maroon" title="c10">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Y</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"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c9">x</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><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">T9</font> = <font color="Maroon" title="c10">y</font> as    <a href="glenum00.html#M1" title="GLENUM00:mode.1">Element</a> of <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="txt" href="#E1:211_1"><span class="lab"><font color="Green" title="E127">A1</font></span></a></span>;<br/><a NAME="E16:211_1_1_1"/>
<font color="Maroon" title="c11">T9</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c7">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:211_1_1_1"><span class="lab"><font color="Green" title="E132">A16</font></span></a>, <a class="ref" href="glib_014.html#T21" onmouseover="rs('glib_014/T21')" onmouseout="rh()">GLIB_014:21</a></span>;<br/><span class="kw">hence </span><a NAME="E17:211_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c9">x</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>; <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="211_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:211_1_1_2"/><span class="lab"><font color="Green" title="E129">A17</font></span>: 
<font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">  ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></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>  &amp; (  for <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive">b<sub>3</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">x</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive" title="b2">y</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><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c3">x</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#K12" title="GLENUM00:func.12">.allTrees()</a>  as    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  ;<br/><span class="kw">take </span><font color="Maroon" title="c4">x</font> = <font color="Maroon" title="c3">x</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="c4">x</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>  &amp; (  for <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Y</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c4">x</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><br/><span class="kw">thus </span><a NAME="E3:211_1_1_2"/>
<font color="Maroon" title="c4">x</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">  for <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Y</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c4">x</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><br/><span class="kw">thus </span><a NAME="E4:211_1_1_2"/>
 for <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">Y</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c4">x</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="txt" href="#E1:211_1_1_2"><span class="lab"><font color="Green" title="E129">A17</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:211"/>
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a>  <a href="orders_1.html#R4" title="ORDERS_1:pred.4">has_upper_Zorn_property_wrt</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="orders_1.html#D10" onmouseover="rs('orders_1/D10')" onmouseout="rh()">ORDERS_1: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>
