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<span class="kw">set </span><font color="Maroon" title="c2">E</font> = <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default">$1 <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default">$1 <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">s</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E2:106_1_1"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span>  &amp; (  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> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b4">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b4">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span>  holds <br/><font color="Maroon" title="c3">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_1.html#S3" onmouseover="rs('funct_1/S3')" onmouseout="rh()">FUNCT_1:sch 3</a>();<br/></span>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="106_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">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c3">s</font> holds <br/><font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">y</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="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c3">s</font> implies <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font> )</span><br/><span class="kw">assume </span><a NAME="E1:106_1_1_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <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"> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:106_1_1_1"/><span class="lab"><font color="Green" title="E47">A2</font></span>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">s</font> &amp; <font color="Maroon" title="c3">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c6">v</font>, <font color="Maroon" title="c7">w</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E5:106_1_1_1"/><span class="lab"><font color="Green" title="E48">A3</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c7">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E6:106_1_1_1"/>
 for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c7">w</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E3:106_1_1_1"><span class="lab"><font color="Green" title="E47">A2</font></span></a></span>;<br/><font color="Maroon" title="c4">y</font> = 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">x</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E3:106_1_1_1"><span class="lab"><font color="Green" title="E47">A2</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c7">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:106_1_1_1"><span class="lab"><font color="Green" title="E48">A3</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E8:106_1_1_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_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"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">s</font> = <font color="Maroon" title="c3">s</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a>, <a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  =  <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p3">(<span class="default">$1 <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon">H<sub>1</sub></font>($1)</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<font color="Maroon">H<sub>1</sub></font>($1), the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p2">(<span class="default">$1 <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon">H<sub>1</sub></font>($1)</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>);<br/>
<a NAME="E5:106_1_1"/><span class="lab"><font color="Green" title="E47">A4</font></span>: 
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Maroon">H<sub>2</sub></font>(<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#D8" onmouseover="rs('funcop_1/D8')" onmouseout="rh()">FUNCOP_1:def 8</a>, <a class="ref" href="glibpre0.html#T3" onmouseover="rs('glibpre0/T3')" onmouseout="rh()">GLIBPRE0:3</a></span>;<br/>
<a NAME="E6:106_1_1"/><span class="lab"><font color="Green" title="E48">A5</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds  <font color="Maroon">H<sub>2</sub></font>(<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="106_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">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">H<sub>2</sub></font>(<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/>

<a NAME="E1:106_1_1_2"/><span class="lab"><font color="Green" title="E49">A6</font></span>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glibpre0.html#T3" onmouseover="rs('glibpre0/T3')" onmouseout="rh()">GLIBPRE0:3</a></span>;<br/>
<span class="kw">thus </span><a NAME="E2:106_1_1_2"/>
<font color="Maroon">H<sub>2</sub></font>(<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_2"><span class="lab"><font color="Green" title="E49">A6</font></span></a>, <a class="ref" href="funcop_1.html#D8" onmouseover="rs('funcop_1/D8')" onmouseout="rh()">FUNCOP_1:def 8</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">consider </span><font color="Maroon" title="c5">t</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E8:106_1_1"/><span class="lab"><font color="Green" title="E49">A7</font></span>: 
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c5">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span>  &amp; (  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> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b4">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b4">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span>  holds <br/><font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">x</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_1.html#S3" onmouseover="rs('funct_1/S3')" onmouseout="rh()">FUNCT_1:sch 3</a>();<br/></span>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="106_1_1_3"><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#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c5">t</font> holds <br/><font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">y</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="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c5">t</font> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font> )</span><br/><span class="kw">assume </span><a NAME="E1:106_1_1_3"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c5">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="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:106_1_1_3"/><span class="lab"><font color="Green" title="E50">A8</font></span>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c5">t</font> &amp; <font color="Maroon" title="c5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c8">v</font>, <font color="Maroon" title="c9">w</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E5:106_1_1_3"/><span class="lab"><font color="Green" title="E51">A9</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c9">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E6:106_1_1_3"/>
 for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c9">w</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E3:106_1_1_3"><span class="lab"><font color="Green" title="E50">A8</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_3_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="E1:106_1_1_3_1"/>
( <font color="Maroon" title="c8">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">w</font> or <font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">w</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:106_1_1_3_1_1"/><span class="lab"><font color="Green" title="E52">A10</font></span>: 
<font color="Maroon" title="c8">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">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="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span><br/><div class="add"><font color="Maroon" title="c6">y</font> = 
<font color="Maroon">H<sub>2</sub></font>(<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c9">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E3:106_1_1_3"><span class="lab"><font color="Green" title="E50">A8</font></span></a>, <a class="txt" href="#E5:106_1_1_3"><span class="lab"><font color="Green" title="E51">A9</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c9">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c9">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:106_1_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="#E1:106_1_1_3_1_1"><span class="lab"><font color="Green" title="E52">A10</font></span></a></span>
;<br/><a NAME="E3:106_1_1_3_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v</font> or <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">w</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_3_1_1"><span class="lab"><font color="Green" title="E52">A10</font></span></a>, <a class="ref" href="glibpre0.html#T4" onmouseover="rs('glibpre0/T4')" onmouseout="rh()">GLIBPRE0:4</a></span>;<br/><span class="kw">hence </span><a NAME="E4:106_1_1_3_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:106_1_1_3_1_2"/><span class="lab"><font color="Green" title="E52">A11</font></span>: 
<font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">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="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span><br/><div class="add"><font color="Maroon" title="c6">y</font> = 
<font color="Maroon">H<sub>2</sub></font>(<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c9">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E3:106_1_1_3"><span class="lab"><font color="Green" title="E50">A8</font></span></a>, <a class="txt" href="#E5:106_1_1_3"><span class="lab"><font color="Green" title="E51">A9</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c8">v</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:106_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E1:106_1_1_3_1_2"><span class="lab"><font color="Green" title="E52">A11</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E3:106_1_1_3_1_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_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"> 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="c6">t</font> = <font color="Maroon" title="c5">t</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a>, <a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">H</font> =  <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>);<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="106_1_1_4"><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">e1</font>, <font color="Olive" title="b2">e2</font>, <font color="Olive" title="b3">v1</font>, <font color="Olive" title="b4">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b3">v1</font>,<font color="Olive" title="b4">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b5">v</font>,<font color="Olive" title="b6">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b5">v</font>, <font color="Olive" title="b6">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b7">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b7">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b5">v</font>,<font color="Olive" title="b6">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) &amp; <font color="Olive" title="b2">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b3">v1</font>,<font color="Olive" title="b4">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b5">v</font>,<font color="Olive" title="b6">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b5">v</font>, <font color="Olive" title="b6">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b7">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b7">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b5">v</font>,<font color="Olive" title="b6">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) holds <br/><font color="Olive" title="b1">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">e2</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">e1</font>, <font color="Maroon" title="c9">e2</font>, <font color="Maroon" title="c10">v1</font>, <font color="Maroon" title="c11">v2</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="c8">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:106_1_1_4"/><span class="lab"><font color="Green" title="E50">A12</font></span>: 
( <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><a NAME="E2:106_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c8">e1</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> <span class="p1">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"> { <span class="default"> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p4"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><a NAME="E3:106_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A13</font></span>: 
<font color="Maroon" title="c8">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> 
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c13">b</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E5:106_1_1_4"/><span class="lab"><font color="Green" title="E52">A14</font></span>: 
<font color="Maroon" title="c8">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E6:106_1_1_4"/>
 for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font>,<font color="Maroon" title="c1">G</font>
 ;<br/><a NAME="E7:106_1_1_4"/>
<font color="Maroon" title="c9">e2</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> <span class="p1">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"> { <span class="default"> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p4"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4"><span class="lab"><font color="Green" title="E50">A12</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><a NAME="E8:106_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A15</font></span>: 
<font color="Maroon" title="c9">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> 
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">c</font>, <font color="Maroon" title="c15">d</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E10:106_1_1_4"/><span class="lab"><font color="Green" title="E54">A16</font></span>: 
<font color="Maroon" title="c9">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E11:106_1_1_4"/>
 for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font>,<font color="Maroon" title="c1">G</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_4_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="E1:106_1_1_4_1"/>
( ( <font color="Maroon" title="c12">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">b</font> &amp; <font color="Maroon" title="c14">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">d</font> ) or ( <font color="Maroon" title="c12">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">b</font> &amp; <font color="Maroon" title="c14">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">d</font> ) or ( <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> &amp; <font color="Maroon" title="c14">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">d</font> ) or ( <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> &amp; <font color="Maroon" title="c14">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">d</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:106_1_1_4_1_1"/><span class="lab"><font color="Green" title="E55">A17</font></span>: 
( <font color="Maroon" title="c12">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">b</font> &amp; <font color="Maroon" title="c14">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">d</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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_4_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="E1:106_1_1_4_1_1_1"/>
( ( <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) ) or ( <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c10">v1</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) ) or ( <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c10">v1</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c10">v1</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) ) or ( <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c10">v1</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4"><span class="lab"><font color="Green" title="E50">A12</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_4_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:106_1_1_4_1_1_1_1"/><span class="lab"><font color="Green" title="E56">A18</font></span>: 
( <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="lab"><font color="Green" title="E57">A19</font></span>: <font color="Maroon" title="c10">v1</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_1_1_1"><span class="lab"><font color="Green" title="E56">A18</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/>.= 
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>

<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E3:106_1_1_4"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>

;<br/><span class="lab"><font color="Green" title="E58">A20</font></span>: <font color="Maroon" title="c11">v2</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_1_1_1"><span class="lab"><font color="Green" title="E56">A18</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/>.= 
<font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>

<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">e1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E3:106_1_1_4"><span class="lab"><font color="Green" title="E51">A13</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:106_1_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="#E1:106_1_1_4_1_1"><span class="lab"><font color="Green" title="E55">A17</font></span></a>, <a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a></span>
;<br/><span class="lab"><font color="Green" title="E59">A21</font></span>: <font color="Maroon" title="c10">v1</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_1_1_1"><span class="lab"><font color="Green" title="E56">A18</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/>.= 
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>

<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E8:106_1_1_4"><span class="lab"><font color="Green" title="E53">A15</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>

;<br/><font color="Maroon" title="c11">v2</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_1_1_1"><span class="lab"><font color="Green" title="E56">A18</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/>.= 
<font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>

<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c9">e2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E8:106_1_1_4"><span class="lab"><font color="Green" title="E53">A15</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:106_1_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="#E1:106_1_1_4_1_1"><span class="lab"><font color="Green" title="E55">A17</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a></span>
;<br/><a NAME="E6:106_1_1_4_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A22</font></span>: 
( ( <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">d</font> ) or ( <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">d</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_1"><span class="lab"><font color="Green" title="E55">A17</font></span></a>, <a class="txt" href="#E4:106_1_1_4_1_1_1_1"><span class="lab"><font color="Green" title="E59">A21</font></span></a>, <a class="ref" href="glibpre0.html#T4" onmouseover="rs('glibpre0/T4')" onmouseout="rh()">GLIBPRE0:4</a></span>;<br/><a NAME="E7:106_1_1_4_1_1_1_1"/>
( ( <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> ) or ( <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_1"><span class="lab"><font color="Green" title="E55">A17</font></span></a>, <a class="txt" href="#E2:106_1_1_4_1_1_1_1"><span class="lab"><font color="Green" title="E57">A19</font></span></a>, <a class="txt" href="#E3:106_1_1_4_1_1_1_1"><span class="lab"><font color="Green" title="E58">A20</font></span></a>, <a class="ref" href="glibpre0.html#T4" onmouseover="rs('glibpre0/T4')" onmouseout="rh()">GLIBPRE0:4</a></span>;<br/><span class="kw">hence </span><a NAME="E8:106_1_1_4_1_1_1_1"/>
<font color="Maroon" title="c8">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a>, <a class="txt" href="#E6:106_1_1_4_1_1_1_1"><span class="lab"><font color="Green" title="E60">A22</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_4_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:106_1_1_4_1_1_1_2"/><span class="lab"><font color="Green" title="E56">A23</font></span>: 
( <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c10">v1</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="lab"><font color="Green" title="E57">A24</font></span>: <font color="Maroon" title="c10">v1</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_1_1_2"><span class="lab"><font color="Green" title="E56">A23</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/>.= 
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>

<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E3:106_1_1_4"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>

;<br/><span class="lab"><font color="Green" title="E58">A25</font></span>: <font color="Maroon" title="c11">v2</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_1_1_2"><span class="lab"><font color="Green" title="E56">A23</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/>.= 
<font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>

<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">e1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E3:106_1_1_4"><span class="lab"><font color="Green" title="E51">A13</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:106_1_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="#E1:106_1_1_4_1_1"><span class="lab"><font color="Green" title="E55">A17</font></span></a>, <a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a></span>
;<br/><span class="lab"><font color="Green" title="E59">A26</font></span>: <font color="Maroon" title="c10">v1</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_1_1_2"><span class="lab"><font color="Green" title="E56">A23</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/>.= 
<font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>

<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c9">e2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E8:106_1_1_4"><span class="lab"><font color="Green" title="E53">A15</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:106_1_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="#E1:106_1_1_4_1_1"><span class="lab"><font color="Green" title="E55">A17</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a></span>
;<br/><font color="Maroon" title="c11">v2</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_1_1_2"><span class="lab"><font color="Green" title="E56">A23</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/>.= 
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>

<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E8:106_1_1_4"><span class="lab"><font color="Green" title="E53">A15</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>

;<br/><a NAME="E6:106_1_1_4_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A27</font></span>: 
( ( <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">d</font> ) or ( <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">d</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_1"><span class="lab"><font color="Green" title="E55">A17</font></span></a>, <a class="txt" href="#E4:106_1_1_4_1_1_1_2"><span class="lab"><font color="Green" title="E59">A26</font></span></a>, <a class="ref" href="glibpre0.html#T4" onmouseover="rs('glibpre0/T4')" onmouseout="rh()">GLIBPRE0:4</a></span>;<br/><a NAME="E7:106_1_1_4_1_1_1_2"/>
( ( <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> ) or ( <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_1"><span class="lab"><font color="Green" title="E55">A17</font></span></a>, <a class="txt" href="#E2:106_1_1_4_1_1_1_2"><span class="lab"><font color="Green" title="E57">A24</font></span></a>, <a class="txt" href="#E3:106_1_1_4_1_1_1_2"><span class="lab"><font color="Green" title="E58">A25</font></span></a>, <a class="ref" href="glibpre0.html#T4" onmouseover="rs('glibpre0/T4')" onmouseout="rh()">GLIBPRE0:4</a></span>;<br/><span class="kw">hence </span><a NAME="E8:106_1_1_4_1_1_1_2"/>
<font color="Maroon" title="c8">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a>, <a class="txt" href="#E6:106_1_1_4_1_1_1_2"><span class="lab"><font color="Green" title="E60">A27</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_4_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:106_1_1_4_1_1_1_3"/><span class="lab"><font color="Green" title="E56">A28</font></span>: 
( <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c10">v1</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c10">v1</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="lab"><font color="Green" title="E57">A29</font></span>: <font color="Maroon" title="c10">v1</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_1_1_3"><span class="lab"><font color="Green" title="E56">A28</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/>.= 
<font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>

<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">e1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E3:106_1_1_4"><span class="lab"><font color="Green" title="E51">A13</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:106_1_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="#E1:106_1_1_4_1_1"><span class="lab"><font color="Green" title="E55">A17</font></span></a>, <a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a></span>
;<br/><span class="lab"><font color="Green" title="E58">A30</font></span>: <font color="Maroon" title="c11">v2</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_1_1_3"><span class="lab"><font color="Green" title="E56">A28</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/>.= 
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>

<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E3:106_1_1_4"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>

;<br/><span class="lab"><font color="Green" title="E59">A31</font></span>: <font color="Maroon" title="c10">v1</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_1_1_3"><span class="lab"><font color="Green" title="E56">A28</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/>.= 
<font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>

<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c9">e2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E8:106_1_1_4"><span class="lab"><font color="Green" title="E53">A15</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:106_1_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="#E1:106_1_1_4_1_1"><span class="lab"><font color="Green" title="E55">A17</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a></span>
;<br/><font color="Maroon" title="c11">v2</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_1_1_3"><span class="lab"><font color="Green" title="E56">A28</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/>.= 
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>

<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E8:106_1_1_4"><span class="lab"><font color="Green" title="E53">A15</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>

;<br/><a NAME="E6:106_1_1_4_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A32</font></span>: 
( ( <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">d</font> ) or ( <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">d</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_1"><span class="lab"><font color="Green" title="E55">A17</font></span></a>, <a class="txt" href="#E4:106_1_1_4_1_1_1_3"><span class="lab"><font color="Green" title="E59">A31</font></span></a>, <a class="ref" href="glibpre0.html#T4" onmouseover="rs('glibpre0/T4')" onmouseout="rh()">GLIBPRE0:4</a></span>;<br/><a NAME="E7:106_1_1_4_1_1_1_3"/>
( ( <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> ) or ( <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_1"><span class="lab"><font color="Green" title="E55">A17</font></span></a>, <a class="txt" href="#E2:106_1_1_4_1_1_1_3"><span class="lab"><font color="Green" title="E57">A29</font></span></a>, <a class="txt" href="#E3:106_1_1_4_1_1_1_3"><span class="lab"><font color="Green" title="E58">A30</font></span></a>, <a class="ref" href="glibpre0.html#T4" onmouseover="rs('glibpre0/T4')" onmouseout="rh()">GLIBPRE0:4</a></span>;<br/><span class="kw">hence </span><a NAME="E8:106_1_1_4_1_1_1_3"/>
<font color="Maroon" title="c8">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a>, <a class="txt" href="#E6:106_1_1_4_1_1_1_3"><span class="lab"><font color="Green" title="E60">A32</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_4_1_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:106_1_1_4_1_1_1_4"/><span class="lab"><font color="Green" title="E56">A33</font></span>: 
( <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c10">v1</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="lab"><font color="Green" title="E57">A34</font></span>: <font color="Maroon" title="c10">v1</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_1_1_4"><span class="lab"><font color="Green" title="E56">A33</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/>.= 
<font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>

<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">e1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E3:106_1_1_4"><span class="lab"><font color="Green" title="E51">A13</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:106_1_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="#E1:106_1_1_4_1_1"><span class="lab"><font color="Green" title="E55">A17</font></span></a>, <a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a></span>
;<br/><span class="lab"><font color="Green" title="E58">A35</font></span>: <font color="Maroon" title="c11">v2</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_1_1_4"><span class="lab"><font color="Green" title="E56">A33</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/>.= 
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>

<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E3:106_1_1_4"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>

;<br/><span class="lab"><font color="Green" title="E59">A36</font></span>: <font color="Maroon" title="c10">v1</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_1_1_4"><span class="lab"><font color="Green" title="E56">A33</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/>.= 
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>

<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E8:106_1_1_4"><span class="lab"><font color="Green" title="E53">A15</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>

;<br/><font color="Maroon" title="c11">v2</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_1_1_4"><span class="lab"><font color="Green" title="E56">A33</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/>.= 
<font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>

<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c9">e2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E8:106_1_1_4"><span class="lab"><font color="Green" title="E53">A15</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:106_1_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="#E1:106_1_1_4_1_1"><span class="lab"><font color="Green" title="E55">A17</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a></span>
;<br/><a NAME="E6:106_1_1_4_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A37</font></span>: 
( ( <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">d</font> ) or ( <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">d</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_1"><span class="lab"><font color="Green" title="E55">A17</font></span></a>, <a class="txt" href="#E4:106_1_1_4_1_1_1_4"><span class="lab"><font color="Green" title="E59">A36</font></span></a>, <a class="ref" href="glibpre0.html#T4" onmouseover="rs('glibpre0/T4')" onmouseout="rh()">GLIBPRE0:4</a></span>;<br/><a NAME="E7:106_1_1_4_1_1_1_4"/>
( ( <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> ) or ( <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_1"><span class="lab"><font color="Green" title="E55">A17</font></span></a>, <a class="txt" href="#E2:106_1_1_4_1_1_1_4"><span class="lab"><font color="Green" title="E57">A34</font></span></a>, <a class="txt" href="#E3:106_1_1_4_1_1_1_4"><span class="lab"><font color="Green" title="E58">A35</font></span></a>, <a class="ref" href="glibpre0.html#T4" onmouseover="rs('glibpre0/T4')" onmouseout="rh()">GLIBPRE0:4</a></span>;<br/><span class="kw">hence </span><a NAME="E8:106_1_1_4_1_1_1_4"/>
<font color="Maroon" title="c8">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a>, <a class="txt" href="#E6:106_1_1_4_1_1_1_4"><span class="lab"><font color="Green" title="E60">A37</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:106_1_1_4_1_2"/><span class="lab"><font color="Green" title="E55">A38</font></span>: 
( <font color="Maroon" title="c12">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">b</font> &amp; <font color="Maroon" title="c14">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">d</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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_4_1_2_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="E1:106_1_1_4_1_2_1"/>
( ( <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) ) or ( <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c10">v1</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) ) or ( <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c10">v1</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c10">v1</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) ) or ( <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c10">v1</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4"><span class="lab"><font color="Green" title="E50">A12</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_4_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:106_1_1_4_1_2_1_1"/><span class="lab"><font color="Green" title="E56">A39</font></span>: 
( <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="lab"><font color="Green" title="E57">A40</font></span>: <font color="Maroon" title="c10">v1</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_2_1_1"><span class="lab"><font color="Green" title="E56">A39</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/>.= 
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>

<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E3:106_1_1_4"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>

;<br/><span class="lab"><font color="Green" title="E58">A41</font></span>: <font color="Maroon" title="c11">v2</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_2_1_1"><span class="lab"><font color="Green" title="E56">A39</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/>.= 
<font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>

<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">e1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E3:106_1_1_4"><span class="lab"><font color="Green" title="E51">A13</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:106_1_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="#E1:106_1_1_4_1_2"><span class="lab"><font color="Green" title="E55">A38</font></span></a>, <a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a></span>
;<br/><span class="lab"><font color="Green" title="E59">A42</font></span>: <font color="Maroon" title="c10">v1</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_2_1_1"><span class="lab"><font color="Green" title="E56">A39</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/>.= 
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>

<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E8:106_1_1_4"><span class="lab"><font color="Green" title="E53">A15</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>

;<br/><span class="lab"><font color="Green" title="E60">A43</font></span>: <font color="Maroon" title="c11">v2</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_2_1_1"><span class="lab"><font color="Green" title="E56">A39</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/>.= 
<font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>

<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c9">e2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E8:106_1_1_4"><span class="lab"><font color="Green" title="E53">A15</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:106_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E1:106_1_1_4_1_2"><span class="lab"><font color="Green" title="E55">A38</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a></span>
;<br/><a NAME="E6:106_1_1_4_1_2_1_1"/>
( ( <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> ) or ( <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_2"><span class="lab"><font color="Green" title="E55">A38</font></span></a>, <a class="txt" href="#E2:106_1_1_4_1_2_1_1"><span class="lab"><font color="Green" title="E57">A40</font></span></a>, <a class="txt" href="#E3:106_1_1_4_1_2_1_1"><span class="lab"><font color="Green" title="E58">A41</font></span></a>, <a class="ref" href="glibpre0.html#T4" onmouseover="rs('glibpre0/T4')" onmouseout="rh()">GLIBPRE0:4</a></span>;<br/><span class="kw">hence </span><a NAME="E7:106_1_1_4_1_2_1_1"/>
<font color="Maroon" title="c8">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_2"><span class="lab"><font color="Green" title="E55">A38</font></span></a>, <a class="txt" href="#E4:106_1_1_4_1_2_1_1"><span class="lab"><font color="Green" title="E59">A42</font></span></a>, <a class="txt" href="#E5:106_1_1_4_1_2_1_1"><span class="lab"><font color="Green" title="E60">A43</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_4_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:106_1_1_4_1_2_1_2"/><span class="lab"><font color="Green" title="E56">A44</font></span>: 
( <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c10">v1</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="lab"><font color="Green" title="E57">A45</font></span>: <font color="Maroon" title="c10">v1</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E56">A44</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/>.= 
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>

<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E3:106_1_1_4"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>

;<br/><span class="lab"><font color="Green" title="E58">A46</font></span>: <font color="Maroon" title="c11">v2</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E56">A44</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/>.= 
<font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>

<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">e1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E3:106_1_1_4"><span class="lab"><font color="Green" title="E51">A13</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:106_1_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="#E1:106_1_1_4_1_2"><span class="lab"><font color="Green" title="E55">A38</font></span></a>, <a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a></span>
;<br/><span class="lab"><font color="Green" title="E59">A47</font></span>: <font color="Maroon" title="c10">v1</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E56">A44</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/>.= 
<font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>

<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c9">e2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E8:106_1_1_4"><span class="lab"><font color="Green" title="E53">A15</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:106_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E1:106_1_1_4_1_2"><span class="lab"><font color="Green" title="E55">A38</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a></span>
;<br/><span class="lab"><font color="Green" title="E60">A48</font></span>: <font color="Maroon" title="c11">v2</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E56">A44</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/>.= 
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>

<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E8:106_1_1_4"><span class="lab"><font color="Green" title="E53">A15</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>

;<br/><a NAME="E6:106_1_1_4_1_2_1_2"/>
( ( <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> ) or ( <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_2"><span class="lab"><font color="Green" title="E55">A38</font></span></a>, <a class="txt" href="#E2:106_1_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E57">A45</font></span></a>, <a class="txt" href="#E3:106_1_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E58">A46</font></span></a>, <a class="ref" href="glibpre0.html#T4" onmouseover="rs('glibpre0/T4')" onmouseout="rh()">GLIBPRE0:4</a></span>;<br/><span class="kw">hence </span><a NAME="E7:106_1_1_4_1_2_1_2"/>
<font color="Maroon" title="c8">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_2"><span class="lab"><font color="Green" title="E55">A38</font></span></a>, <a class="txt" href="#E4:106_1_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E59">A47</font></span></a>, <a class="txt" href="#E5:106_1_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E60">A48</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_4_1_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:106_1_1_4_1_2_1_3"/><span class="lab"><font color="Green" title="E56">A49</font></span>: 
( <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c10">v1</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c10">v1</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="lab"><font color="Green" title="E57">A50</font></span>: <font color="Maroon" title="c10">v1</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_2_1_3"><span class="lab"><font color="Green" title="E56">A49</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/>.= 
<font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>

<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">e1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E3:106_1_1_4"><span class="lab"><font color="Green" title="E51">A13</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:106_1_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="#E1:106_1_1_4_1_2"><span class="lab"><font color="Green" title="E55">A38</font></span></a>, <a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a></span>
;<br/><span class="lab"><font color="Green" title="E58">A51</font></span>: <font color="Maroon" title="c11">v2</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_2_1_3"><span class="lab"><font color="Green" title="E56">A49</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/>.= 
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>

<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E3:106_1_1_4"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>

;<br/><span class="lab"><font color="Green" title="E59">A52</font></span>: <font color="Maroon" title="c10">v1</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_2_1_3"><span class="lab"><font color="Green" title="E56">A49</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/>.= 
<font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>

<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c9">e2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E8:106_1_1_4"><span class="lab"><font color="Green" title="E53">A15</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:106_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E1:106_1_1_4_1_2"><span class="lab"><font color="Green" title="E55">A38</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a></span>
;<br/><span class="lab"><font color="Green" title="E60">A53</font></span>: <font color="Maroon" title="c11">v2</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_2_1_3"><span class="lab"><font color="Green" title="E56">A49</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/>.= 
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>

<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E8:106_1_1_4"><span class="lab"><font color="Green" title="E53">A15</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>

;<br/><a NAME="E6:106_1_1_4_1_2_1_3"/>
( ( <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> ) or ( <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_2"><span class="lab"><font color="Green" title="E55">A38</font></span></a>, <a class="txt" href="#E2:106_1_1_4_1_2_1_3"><span class="lab"><font color="Green" title="E57">A50</font></span></a>, <a class="txt" href="#E3:106_1_1_4_1_2_1_3"><span class="lab"><font color="Green" title="E58">A51</font></span></a>, <a class="ref" href="glibpre0.html#T4" onmouseover="rs('glibpre0/T4')" onmouseout="rh()">GLIBPRE0:4</a></span>;<br/><span class="kw">hence </span><a NAME="E7:106_1_1_4_1_2_1_3"/>
<font color="Maroon" title="c8">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_2"><span class="lab"><font color="Green" title="E55">A38</font></span></a>, <a class="txt" href="#E4:106_1_1_4_1_2_1_3"><span class="lab"><font color="Green" title="E59">A52</font></span></a>, <a class="txt" href="#E5:106_1_1_4_1_2_1_3"><span class="lab"><font color="Green" title="E60">A53</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_4_1_2_1_4"><span class="kw">suppose </span></a><a NAME="E1:106_1_1_4_1_2_1_4"/><span class="lab"><font color="Green" title="E56">A54</font></span>: 
( <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c10">v1</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="lab"><font color="Green" title="E57">A55</font></span>: <font color="Maroon" title="c10">v1</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_2_1_4"><span class="lab"><font color="Green" title="E56">A54</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/>.= 
<font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>

<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">e1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E3:106_1_1_4"><span class="lab"><font color="Green" title="E51">A13</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:106_1_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="#E1:106_1_1_4_1_2"><span class="lab"><font color="Green" title="E55">A38</font></span></a>, <a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a></span>
;<br/><span class="lab"><font color="Green" title="E58">A56</font></span>: <font color="Maroon" title="c11">v2</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_2_1_4"><span class="lab"><font color="Green" title="E56">A54</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/>.= 
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>

<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E3:106_1_1_4"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>

;<br/><span class="lab"><font color="Green" title="E59">A57</font></span>: <font color="Maroon" title="c10">v1</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_2_1_4"><span class="lab"><font color="Green" title="E56">A54</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/>.= 
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>

<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E8:106_1_1_4"><span class="lab"><font color="Green" title="E53">A15</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>

;<br/><span class="lab"><font color="Green" title="E60">A58</font></span>: <font color="Maroon" title="c11">v2</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_2_1_4"><span class="lab"><font color="Green" title="E56">A54</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/>.= 
<font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>

<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c9">e2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E8:106_1_1_4"><span class="lab"><font color="Green" title="E53">A15</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:106_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E1:106_1_1_4_1_2"><span class="lab"><font color="Green" title="E55">A38</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a></span>
;<br/><a NAME="E6:106_1_1_4_1_2_1_4"/>
( ( <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> ) or ( <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_2"><span class="lab"><font color="Green" title="E55">A38</font></span></a>, <a class="txt" href="#E2:106_1_1_4_1_2_1_4"><span class="lab"><font color="Green" title="E57">A55</font></span></a>, <a class="txt" href="#E3:106_1_1_4_1_2_1_4"><span class="lab"><font color="Green" title="E58">A56</font></span></a>, <a class="ref" href="glibpre0.html#T4" onmouseover="rs('glibpre0/T4')" onmouseout="rh()">GLIBPRE0:4</a></span>;<br/><span class="kw">hence </span><a NAME="E7:106_1_1_4_1_2_1_4"/>
<font color="Maroon" title="c8">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_2"><span class="lab"><font color="Green" title="E55">A38</font></span></a>, <a class="txt" href="#E4:106_1_1_4_1_2_1_4"><span class="lab"><font color="Green" title="E59">A57</font></span></a>, <a class="txt" href="#E5:106_1_1_4_1_2_1_4"><span class="lab"><font color="Green" title="E60">A58</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_4_1_3"><span class="kw">suppose </span></a><a NAME="E1:106_1_1_4_1_3"/><span class="lab"><font color="Green" title="E55">A59</font></span>: 
( <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> &amp; <font color="Maroon" title="c14">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">d</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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_4_1_3_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="E1:106_1_1_4_1_3_1"/>
( ( <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) ) or ( <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c10">v1</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) ) or ( <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c10">v1</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c10">v1</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) ) or ( <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c10">v1</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4"><span class="lab"><font color="Green" title="E50">A12</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_4_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:106_1_1_4_1_3_1_1"/><span class="lab"><font color="Green" title="E56">A60</font></span>: 
( <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="lab"><font color="Green" title="E57">A61</font></span>: <font color="Maroon" title="c10">v1</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_3_1_1"><span class="lab"><font color="Green" title="E56">A60</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/>.= 
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>

<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E3:106_1_1_4"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>

;<br/><span class="lab"><font color="Green" title="E58">A62</font></span>: <font color="Maroon" title="c11">v2</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_3_1_1"><span class="lab"><font color="Green" title="E56">A60</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/>.= 
<font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>

<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">e1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E3:106_1_1_4"><span class="lab"><font color="Green" title="E51">A13</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:106_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E1:106_1_1_4_1_3"><span class="lab"><font color="Green" title="E55">A59</font></span></a>, <a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a></span>
;<br/><span class="lab"><font color="Green" title="E59">A63</font></span>: <font color="Maroon" title="c10">v1</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_3_1_1"><span class="lab"><font color="Green" title="E56">A60</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/>.= 
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>

<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E8:106_1_1_4"><span class="lab"><font color="Green" title="E53">A15</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>

;<br/><font color="Maroon" title="c11">v2</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_3_1_1"><span class="lab"><font color="Green" title="E56">A60</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/>.= 
<font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>

<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c9">e2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E8:106_1_1_4"><span class="lab"><font color="Green" title="E53">A15</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:106_1_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="#E1:106_1_1_4_1_3"><span class="lab"><font color="Green" title="E55">A59</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a></span>
;<br/><a NAME="E6:106_1_1_4_1_3_1_1"/><span class="kw">then </span>
( ( <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">d</font> ) or ( <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">d</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_3"><span class="lab"><font color="Green" title="E55">A59</font></span></a>, <a class="txt" href="#E4:106_1_1_4_1_3_1_1"><span class="lab"><font color="Green" title="E59">A63</font></span></a>, <a class="ref" href="glibpre0.html#T4" onmouseover="rs('glibpre0/T4')" onmouseout="rh()">GLIBPRE0:4</a></span>;<br/><span class="kw">hence </span><a NAME="E7:106_1_1_4_1_3_1_1"/>
<font color="Maroon" title="c8">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_3"><span class="lab"><font color="Green" title="E55">A59</font></span></a>, <a class="txt" href="#E2:106_1_1_4_1_3_1_1"><span class="lab"><font color="Green" title="E57">A61</font></span></a>, <a class="txt" href="#E3:106_1_1_4_1_3_1_1"><span class="lab"><font color="Green" title="E58">A62</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_4_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:106_1_1_4_1_3_1_2"/><span class="lab"><font color="Green" title="E56">A64</font></span>: 
( <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c10">v1</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="lab"><font color="Green" title="E57">A65</font></span>: <font color="Maroon" title="c10">v1</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_3_1_2"><span class="lab"><font color="Green" title="E56">A64</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/>.= 
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>

<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E3:106_1_1_4"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>

;<br/><span class="lab"><font color="Green" title="E58">A66</font></span>: <font color="Maroon" title="c11">v2</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_3_1_2"><span class="lab"><font color="Green" title="E56">A64</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/>.= 
<font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>

<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">e1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E3:106_1_1_4"><span class="lab"><font color="Green" title="E51">A13</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:106_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E1:106_1_1_4_1_3"><span class="lab"><font color="Green" title="E55">A59</font></span></a>, <a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a></span>
;<br/><span class="lab"><font color="Green" title="E59">A67</font></span>: <font color="Maroon" title="c10">v1</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_3_1_2"><span class="lab"><font color="Green" title="E56">A64</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/>.= 
<font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>

<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c9">e2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E8:106_1_1_4"><span class="lab"><font color="Green" title="E53">A15</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:106_1_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="#E1:106_1_1_4_1_3"><span class="lab"><font color="Green" title="E55">A59</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a></span>
;<br/><font color="Maroon" title="c11">v2</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_3_1_2"><span class="lab"><font color="Green" title="E56">A64</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/>.= 
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>

<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E8:106_1_1_4"><span class="lab"><font color="Green" title="E53">A15</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>

;<br/><a NAME="E6:106_1_1_4_1_3_1_2"/><span class="kw">then </span>
( ( <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">d</font> ) or ( <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">d</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_3"><span class="lab"><font color="Green" title="E55">A59</font></span></a>, <a class="txt" href="#E4:106_1_1_4_1_3_1_2"><span class="lab"><font color="Green" title="E59">A67</font></span></a>, <a class="ref" href="glibpre0.html#T4" onmouseover="rs('glibpre0/T4')" onmouseout="rh()">GLIBPRE0:4</a></span>;<br/><span class="kw">hence </span><a NAME="E7:106_1_1_4_1_3_1_2"/>
<font color="Maroon" title="c8">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_3"><span class="lab"><font color="Green" title="E55">A59</font></span></a>, <a class="txt" href="#E2:106_1_1_4_1_3_1_2"><span class="lab"><font color="Green" title="E57">A65</font></span></a>, <a class="txt" href="#E3:106_1_1_4_1_3_1_2"><span class="lab"><font color="Green" title="E58">A66</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_4_1_3_1_3"><span class="kw">suppose </span></a><a NAME="E1:106_1_1_4_1_3_1_3"/><span class="lab"><font color="Green" title="E56">A68</font></span>: 
( <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c10">v1</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c10">v1</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="lab"><font color="Green" title="E57">A69</font></span>: <font color="Maroon" title="c10">v1</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_3_1_3"><span class="lab"><font color="Green" title="E56">A68</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/>.= 
<font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>

<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">e1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E3:106_1_1_4"><span class="lab"><font color="Green" title="E51">A13</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:106_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E1:106_1_1_4_1_3"><span class="lab"><font color="Green" title="E55">A59</font></span></a>, <a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a></span>
;<br/><span class="lab"><font color="Green" title="E58">A70</font></span>: <font color="Maroon" title="c11">v2</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_3_1_3"><span class="lab"><font color="Green" title="E56">A68</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/>.= 
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>

<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E3:106_1_1_4"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>

;<br/><span class="lab"><font color="Green" title="E59">A71</font></span>: <font color="Maroon" title="c10">v1</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_3_1_3"><span class="lab"><font color="Green" title="E56">A68</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/>.= 
<font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>

<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c9">e2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E8:106_1_1_4"><span class="lab"><font color="Green" title="E53">A15</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:106_1_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="#E1:106_1_1_4_1_3"><span class="lab"><font color="Green" title="E55">A59</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a></span>
;<br/><font color="Maroon" title="c11">v2</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_3_1_3"><span class="lab"><font color="Green" title="E56">A68</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/>.= 
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>

<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E8:106_1_1_4"><span class="lab"><font color="Green" title="E53">A15</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>

;<br/><a NAME="E6:106_1_1_4_1_3_1_3"/><span class="kw">then </span>
( ( <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">d</font> ) or ( <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">d</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_3"><span class="lab"><font color="Green" title="E55">A59</font></span></a>, <a class="txt" href="#E4:106_1_1_4_1_3_1_3"><span class="lab"><font color="Green" title="E59">A71</font></span></a>, <a class="ref" href="glibpre0.html#T4" onmouseover="rs('glibpre0/T4')" onmouseout="rh()">GLIBPRE0:4</a></span>;<br/><span class="kw">hence </span><a NAME="E7:106_1_1_4_1_3_1_3"/>
<font color="Maroon" title="c8">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_3"><span class="lab"><font color="Green" title="E55">A59</font></span></a>, <a class="txt" href="#E2:106_1_1_4_1_3_1_3"><span class="lab"><font color="Green" title="E57">A69</font></span></a>, <a class="txt" href="#E3:106_1_1_4_1_3_1_3"><span class="lab"><font color="Green" title="E58">A70</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_4_1_3_1_4"><span class="kw">suppose </span></a><a NAME="E1:106_1_1_4_1_3_1_4"/><span class="lab"><font color="Green" title="E56">A72</font></span>: 
( <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c10">v1</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="lab"><font color="Green" title="E57">A73</font></span>: <font color="Maroon" title="c10">v1</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_3_1_4"><span class="lab"><font color="Green" title="E56">A72</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/>.= 
<font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>

<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">e1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E3:106_1_1_4"><span class="lab"><font color="Green" title="E51">A13</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:106_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E1:106_1_1_4_1_3"><span class="lab"><font color="Green" title="E55">A59</font></span></a>, <a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a></span>
;<br/><span class="lab"><font color="Green" title="E58">A74</font></span>: <font color="Maroon" title="c11">v2</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_3_1_4"><span class="lab"><font color="Green" title="E56">A72</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/>.= 
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>

<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E3:106_1_1_4"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>

;<br/><span class="lab"><font color="Green" title="E59">A75</font></span>: <font color="Maroon" title="c10">v1</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_3_1_4"><span class="lab"><font color="Green" title="E56">A72</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/>.= 
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>

<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E8:106_1_1_4"><span class="lab"><font color="Green" title="E53">A15</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>

;<br/><font color="Maroon" title="c11">v2</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_3_1_4"><span class="lab"><font color="Green" title="E56">A72</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/>.= 
<font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>

<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c9">e2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E8:106_1_1_4"><span class="lab"><font color="Green" title="E53">A15</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:106_1_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="#E1:106_1_1_4_1_3"><span class="lab"><font color="Green" title="E55">A59</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a></span>
;<br/><a NAME="E6:106_1_1_4_1_3_1_4"/><span class="kw">then </span>
( ( <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">d</font> ) or ( <font color="Maroon" title="c10">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">d</font> &amp; <font color="Maroon" title="c11">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">c</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_3"><span class="lab"><font color="Green" title="E55">A59</font></span></a>, <a class="txt" href="#E4:106_1_1_4_1_3_1_4"><span class="lab"><font color="Green" title="E59">A75</font></span></a>, <a class="ref" href="glibpre0.html#T4" onmouseover="rs('glibpre0/T4')" onmouseout="rh()">GLIBPRE0:4</a></span>;<br/><span class="kw">hence </span><a NAME="E7:106_1_1_4_1_3_1_4"/>
<font color="Maroon" title="c8">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_3"><span class="lab"><font color="Green" title="E55">A59</font></span></a>, <a class="txt" href="#E2:106_1_1_4_1_3_1_4"><span class="lab"><font color="Green" title="E57">A73</font></span></a>, <a class="txt" href="#E3:106_1_1_4_1_3_1_4"><span class="lab"><font color="Green" title="E58">A74</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_4_1_4"><span class="kw">suppose </span></a><a NAME="E1:106_1_1_4_1_4"/><span class="lab"><font color="Green" title="E55">A76</font></span>: 
( <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> &amp; <font color="Maroon" title="c14">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">d</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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><a NAME="E2:106_1_1_4_1_4"/><span class="lab"><font color="Green" title="E56">A77</font></span>: 
( <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c12">a</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c12">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c14">c</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c14">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T29" onmouseover="rs('enumset1/T29')" onmouseout="rh()">ENUMSET1:29</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_4_1_4_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="E1:106_1_1_4_1_4_1"/>
( ( <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) ) or ( <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c10">v1</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) ) or ( <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c10">v1</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c10">v1</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) ) or ( <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c10">v1</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4"><span class="lab"><font color="Green" title="E50">A12</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_4_1_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:106_1_1_4_1_4_1_1"/><span class="lab"><font color="Green" title="E57">A78</font></span>: 
( <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="lab"><font color="Green" title="E58">A79</font></span>: <font color="Maroon" title="c10">v1</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_4_1_1"><span class="lab"><font color="Green" title="E57">A78</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/>.= 
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>

<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E3:106_1_1_4"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c12">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_4"><span class="lab"><font color="Green" title="E55">A76</font></span></a>, <a class="txt" href="#E2:106_1_1_4_1_4"><span class="lab"><font color="Green" title="E56">A77</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>
;<br/><font color="Maroon" title="c10">v1</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_4_1_1"><span class="lab"><font color="Green" title="E57">A78</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/>.= 
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>

<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E8:106_1_1_4"><span class="lab"><font color="Green" title="E53">A15</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c14">c</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_4"><span class="lab"><font color="Green" title="E55">A76</font></span></a>, <a class="txt" href="#E2:106_1_1_4_1_4"><span class="lab"><font color="Green" title="E56">A77</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:106_1_1_4_1_4_1_1"/>
<font color="Maroon" title="c8">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a>, <a class="txt" href="#E1:106_1_1_4_1_4"><span class="lab"><font color="Green" title="E55">A76</font></span></a>, <a class="txt" href="#E2:106_1_1_4_1_4_1_1"><span class="lab"><font color="Green" title="E58">A79</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_4_1_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:106_1_1_4_1_4_1_2"/><span class="lab"><font color="Green" title="E57">A80</font></span>: 
( <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c10">v1</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="lab"><font color="Green" title="E58">A81</font></span>: <font color="Maroon" title="c10">v1</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_4_1_2"><span class="lab"><font color="Green" title="E57">A80</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/>.= 
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>

<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E3:106_1_1_4"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c12">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_4"><span class="lab"><font color="Green" title="E55">A76</font></span></a>, <a class="txt" href="#E2:106_1_1_4_1_4"><span class="lab"><font color="Green" title="E56">A77</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>
;<br/><font color="Maroon" title="c10">v1</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_4_1_2"><span class="lab"><font color="Green" title="E57">A80</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/>.= 
<font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>

<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c9">e2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E8:106_1_1_4"><span class="lab"><font color="Green" title="E53">A15</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:106_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E1:106_1_1_4_1_4"><span class="lab"><font color="Green" title="E55">A76</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c14">c</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_4"><span class="lab"><font color="Green" title="E55">A76</font></span></a>, <a class="txt" href="#E2:106_1_1_4_1_4"><span class="lab"><font color="Green" title="E56">A77</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:106_1_1_4_1_4_1_2"/>
<font color="Maroon" title="c8">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a>, <a class="txt" href="#E1:106_1_1_4_1_4"><span class="lab"><font color="Green" title="E55">A76</font></span></a>, <a class="txt" href="#E2:106_1_1_4_1_4_1_2"><span class="lab"><font color="Green" title="E58">A81</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_4_1_4_1_3"><span class="kw">suppose </span></a><a NAME="E1:106_1_1_4_1_4_1_3"/><span class="lab"><font color="Green" title="E57">A82</font></span>: 
( <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c10">v1</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c10">v1</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="lab"><font color="Green" title="E58">A83</font></span>: <font color="Maroon" title="c10">v1</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_4_1_3"><span class="lab"><font color="Green" title="E57">A82</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/>.= 
<font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>

<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">e1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E3:106_1_1_4"><span class="lab"><font color="Green" title="E51">A13</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:106_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E1:106_1_1_4_1_4"><span class="lab"><font color="Green" title="E55">A76</font></span></a>, <a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c12">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_4"><span class="lab"><font color="Green" title="E55">A76</font></span></a>, <a class="txt" href="#E2:106_1_1_4_1_4"><span class="lab"><font color="Green" title="E56">A77</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>
;<br/><font color="Maroon" title="c10">v1</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_4_1_3"><span class="lab"><font color="Green" title="E57">A82</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/>.= 
<font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>

<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c9">e2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E8:106_1_1_4"><span class="lab"><font color="Green" title="E53">A15</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:106_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E1:106_1_1_4_1_4"><span class="lab"><font color="Green" title="E55">A76</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c14">c</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_4"><span class="lab"><font color="Green" title="E55">A76</font></span></a>, <a class="txt" href="#E2:106_1_1_4_1_4"><span class="lab"><font color="Green" title="E56">A77</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:106_1_1_4_1_4_1_3"/>
<font color="Maroon" title="c8">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a>, <a class="txt" href="#E1:106_1_1_4_1_4"><span class="lab"><font color="Green" title="E55">A76</font></span></a>, <a class="txt" href="#E2:106_1_1_4_1_4_1_3"><span class="lab"><font color="Green" title="E58">A83</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_4_1_4_1_4"><span class="kw">suppose </span></a><a NAME="E1:106_1_1_4_1_4_1_4"/><span class="lab"><font color="Green" title="E57">A84</font></span>: 
( <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c11">v2</font>,<font color="Maroon" title="c10">v1</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) &amp; <font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">v2</font>, <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="lab"><font color="Green" title="E58">A85</font></span>: <font color="Maroon" title="c10">v1</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_4_1_4"><span class="lab"><font color="Green" title="E57">A84</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/>.= 
<font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e1</font>

<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">e1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E3:106_1_1_4"><span class="lab"><font color="Green" title="E51">A13</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c12">a</font>,<font color="Maroon" title="c13">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:106_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E1:106_1_1_4_1_4"><span class="lab"><font color="Green" title="E55">A76</font></span></a>, <a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c12">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_4"><span class="lab"><font color="Green" title="E55">A76</font></span></a>, <a class="txt" href="#E2:106_1_1_4_1_4"><span class="lab"><font color="Green" title="E56">A77</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>
;<br/><font color="Maroon" title="c10">v1</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"> { <span class="default"> <span class="p4"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p5"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_4_1_4"><span class="lab"><font color="Green" title="E57">A84</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/>.= 
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e2</font>

<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">c</font>,<font color="Maroon" title="c15">d</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E8:106_1_1_4"><span class="lab"><font color="Green" title="E53">A15</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c14">c</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_4_1_4"><span class="lab"><font color="Green" title="E55">A76</font></span></a>, <a class="txt" href="#E2:106_1_1_4_1_4"><span class="lab"><font color="Green" title="E56">A77</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:106_1_1_4_1_4_1_4"/>
<font color="Maroon" title="c8">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:106_1_1_4"><span class="lab"><font color="Green" title="E52">A14</font></span></a>, <a class="txt" href="#E10:106_1_1_4"><span class="lab"><font color="Green" title="E54">A16</font></span></a>, <a class="txt" href="#E1:106_1_1_4_1_4"><span class="lab"><font color="Green" title="E55">A76</font></span></a>, <a class="txt" href="#E2:106_1_1_4_1_4_1_4"><span class="lab"><font color="Green" title="E58">A85</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></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="c8">H</font> =  <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> (<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> ,<font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c6">t</font>) as   <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</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_000.html#D20" onmouseover="rs('glib_000/D20')" onmouseout="rh()">GLIB_000:def 20</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">H</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"> (  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c8">H</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="c1">G</font> &amp;  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c8">H</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> &amp; (  for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> holds <br/> (  ex <font color="Olive" title="b3">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b3">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> iff  for <font color="Olive" title="b3">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c8">H</font> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E13:106_1_1"/>
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c8">H</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="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"> (  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c8">H</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> &amp; (  for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> holds <br/> (  ex <font color="Olive" title="b3">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b3">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> iff  for <font color="Olive" title="b3">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c8">H</font> ) ) )</span><br/>

<a NAME="E14:106_1_1"/>
<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span>  <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="106_1_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:106_1_1_5"/>
<span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span>  <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <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="c9">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:106_1_1_5"/><span class="lab"><font color="Green" title="E50">A86</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span>  <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>
 <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/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">x</font> = <font color="Maroon" title="c9">x</font> as    <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="E5:106_1_1_5"/>
<font color="Maroon" title="c10">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:106_1_1_5"><span class="lab"><font color="Green" title="E50">A86</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">v</font>, <font color="Maroon" title="c12">w</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E7:106_1_1_5"/><span class="lab"><font color="Green" title="E51">A87</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">v</font>,<font color="Maroon" title="c12">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E8:106_1_1_5"/>
 for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c11">v</font>,<font color="Maroon" title="c12">w</font>,<font color="Maroon" title="c1">G</font>
 ;<br/>
<a NAME="E9:106_1_1_5"/><span class="lab"><font color="Green" title="E52">A88</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="tarski.html#R2" title="TARSKI: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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:106_1_1_5"><span class="lab"><font color="Green" title="E50">A86</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E10:106_1_1_5"/><span class="lab"><font color="Green" title="E53">A89</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="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E11:106_1_1_5"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c11">v</font>,<font color="Maroon" title="c12">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:106_1_1_5"><span class="lab"><font color="Green" title="E51">A87</font></span></a>, <a class="ref" href="tarski.html#D5" onmouseover="rs('tarski/D5')" onmouseout="rh()">TARSKI:def 5</a></span>;<br/>
<a NAME="E12:106_1_1_5"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:106_1_1_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:106_1_1_5"><span class="lab"><font color="Green" title="E52">A88</font></span></a>, <a class="txt" href="#E10:106_1_1_5"><span class="lab"><font color="Green" title="E53">A89</font></span></a>, <a class="ref" href="xregular.html#T7" onmouseover="rs('xregular/T7')" onmouseout="rh()">XREGULAR:7</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="E15:106_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c8">H</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E16:106_1_1"/>
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c8">H</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</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="ref" href="xboole_0.html#D7" onmouseover="rs('xboole_0/D7')" onmouseout="rh()">XBOOLE_0:def 7</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">v</font>, <font color="Olive" title="b2">w</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> holds <br/> (  ex <font color="Olive" title="b3">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b3">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> iff  for <font color="Olive" title="b3">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c8">H</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c9">v</font>, <font color="Maroon" title="c10">w</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <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"> (  ex <font color="Olive" title="b1">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c1">G</font> iff  for <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c8">H</font> )</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"> ( (  for <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c8">H</font> ) implies  ex <font color="Olive" title="b1">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c1">G</font> )</span>
<div class="add"><span class="kw">given </span><font color="Maroon" title="c11">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><a NAME="E2:106_1_1_6"/><span class="lab"><font color="Green" title="E50">A90</font></span>: 
<font color="Maroon" title="c11">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font>,<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">  for <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c8">H</font></span><br/><span class="kw">given </span><font color="Maroon" title="c12">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><a NAME="E4:106_1_1_6"/><span class="lab"><font color="Green" title="E51">A91</font></span>: 
<font color="Maroon" title="c12">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c8">H</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"> contradiction</span><br/><a NAME="E5:106_1_1_6"/>
<font color="Maroon" title="c12">e2</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="c8">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:106_1_1_6"><span class="lab"><font color="Green" title="E51">A91</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><a NAME="E6:106_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A92</font></span>: 
<font color="Maroon" title="c12">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> 
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">a</font>, <font color="Maroon" title="c14">b</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E8:106_1_1_6"/><span class="lab"><font color="Green" title="E53">A93</font></span>: 
<font color="Maroon" title="c12">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c14">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E9:106_1_1_6"/><span class="lab"><font color="Green" title="E54">A94</font></span>: 
 for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c14">b</font>,<font color="Maroon" title="c1">G</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_6_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="E1:106_1_1_6_1"/>
( <font color="Maroon" title="c13">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">b</font> or <font color="Maroon" title="c13">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_6_1_1"><span class="kw">suppose </span></a><a NAME="E1:106_1_1_6_1_1"/><span class="lab"><font color="Green" title="E55">A95</font></span>: 
<font color="Maroon" title="c13">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">b</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"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_6_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="E1:106_1_1_6_1_1_1"/>
( <font color="Maroon" title="c12">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c8">H</font> or <font color="Maroon" title="c12">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c8">H</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:106_1_1_6"><span class="lab"><font color="Green" title="E51">A91</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_6_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:106_1_1_6_1_1_1_1"/><span class="lab"><font color="Green" title="E56">A96</font></span>: 
<font color="Maroon" title="c12">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c8">H</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"> contradiction</span><br/><div class="add"><span class="lab"><font color="Green" title="E57">A97</font></span>: <font color="Maroon" title="c9">v</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c8">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_6_1_1_1_1"><span class="lab"><font color="Green" title="E56">A96</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/>.= 
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font>

<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c14">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E6:106_1_1_6"><span class="lab"><font color="Green" title="E52">A92</font></span></a>, <a class="txt" href="#E8:106_1_1_6"><span class="lab"><font color="Green" title="E53">A93</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c14">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>

;<br/><font color="Maroon" title="c10">w</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c8">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_6_1_1_1_1"><span class="lab"><font color="Green" title="E56">A96</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/>.= 
<font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font>

<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c12">e2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E6:106_1_1_6"><span class="lab"><font color="Green" title="E52">A92</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c14">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c14">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:106_1_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="#E8:106_1_1_6"><span class="lab"><font color="Green" title="E53">A93</font></span></a>, <a class="txt" href="#E1:106_1_1_6_1_1"><span class="lab"><font color="Green" title="E55">A95</font></span></a></span>
;<br/><a NAME="E4:106_1_1_6_1_1_1_1"/><span class="kw">then </span>
( ( <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a</font> &amp; <font color="Maroon" title="c10">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">b</font> ) or ( <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">b</font> &amp; <font color="Maroon" title="c10">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_6_1_1"><span class="lab"><font color="Green" title="E55">A95</font></span></a>, <a class="txt" href="#E2:106_1_1_6_1_1_1_1"><span class="lab"><font color="Green" title="E57">A97</font></span></a>, <a class="ref" href="glibpre0.html#T4" onmouseover="rs('glibpre0/T4')" onmouseout="rh()">GLIBPRE0:4</a></span>;<br/><span class="kw">hence </span><a NAME="E5:106_1_1_6_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1_6"><span class="lab"><font color="Green" title="E50">A90</font></span></a>, <a class="txt" href="#E9:106_1_1_6"><span class="lab"><font color="Green" title="E54">A94</font></span></a>, <a class="ref" href="glib_000.html#T14" onmouseover="rs('glib_000/T14')" onmouseout="rh()">GLIB_000:14</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="106_1_1_6_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:106_1_1_6_1_1_1_2"/><span class="lab"><font color="Green" title="E56">A98</font></span>: 
<font color="Maroon" title="c12">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c8">H</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"> contradiction</span><br/><div class="add"><span class="lab"><font color="Green" title="E57">A99</font></span>: <font color="Maroon" title="c9">v</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c8">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_6_1_1_1_2"><span class="lab"><font color="Green" title="E56">A98</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/>.= 
<font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font>

<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c12">e2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E6:106_1_1_6"><span class="lab"><font color="Green" title="E52">A92</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c14">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c14">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:106_1_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="#E8:106_1_1_6"><span class="lab"><font color="Green" title="E53">A93</font></span></a>, <a class="txt" href="#E1:106_1_1_6_1_1"><span class="lab"><font color="Green" title="E55">A95</font></span></a></span>
;<br/><font color="Maroon" title="c10">w</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c8">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_6_1_1_1_2"><span class="lab"><font color="Green" title="E56">A98</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/>.= 
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font>

<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c14">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E6:106_1_1_6"><span class="lab"><font color="Green" title="E52">A92</font></span></a>, <a class="txt" href="#E8:106_1_1_6"><span class="lab"><font color="Green" title="E53">A93</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c14">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>

;<br/><a NAME="E4:106_1_1_6_1_1_1_2"/><span class="kw">then </span>
( ( <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a</font> &amp; <font color="Maroon" title="c10">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">b</font> ) or ( <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">b</font> &amp; <font color="Maroon" title="c10">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_6_1_1"><span class="lab"><font color="Green" title="E55">A95</font></span></a>, <a class="txt" href="#E2:106_1_1_6_1_1_1_2"><span class="lab"><font color="Green" title="E57">A99</font></span></a>, <a class="ref" href="glibpre0.html#T4" onmouseover="rs('glibpre0/T4')" onmouseout="rh()">GLIBPRE0:4</a></span>;<br/><span class="kw">hence </span><a NAME="E5:106_1_1_6_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1_6"><span class="lab"><font color="Green" title="E50">A90</font></span></a>, <a class="txt" href="#E9:106_1_1_6"><span class="lab"><font color="Green" title="E54">A94</font></span></a>, <a class="ref" href="glib_000.html#T14" onmouseover="rs('glib_000/T14')" onmouseout="rh()">GLIB_000:14</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_6_1_2"><span class="kw">suppose </span></a><a NAME="E1:106_1_1_6_1_2"/><span class="lab"><font color="Green" title="E55">A100</font></span>: 
<font color="Maroon" title="c13">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">b</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"> contradiction</span><br/><div class="add"><a NAME="E2:106_1_1_6_1_2"/><span class="lab"><font color="Green" title="E56">A101</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c13">a</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c13">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T29" onmouseover="rs('enumset1/T29')" onmouseout="rh()">ENUMSET1:29</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_6_1_2_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="E1:106_1_1_6_1_2_1"/>
( <font color="Maroon" title="c12">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c8">H</font> or <font color="Maroon" title="c12">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c8">H</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:106_1_1_6"><span class="lab"><font color="Green" title="E51">A91</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_6_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:106_1_1_6_1_2_1_1"/><span class="lab"><font color="Green" title="E57">A102</font></span>: 
<font color="Maroon" title="c12">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c8">H</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"> contradiction</span><br/><div class="add"><span class="lab"><font color="Green" title="E58">A103</font></span>: <font color="Maroon" title="c9">v</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c8">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_6_1_2_1_1"><span class="lab"><font color="Green" title="E57">A102</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/>.= 
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font>

<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c14">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E6:106_1_1_6"><span class="lab"><font color="Green" title="E52">A92</font></span></a>, <a class="txt" href="#E8:106_1_1_6"><span class="lab"><font color="Green" title="E53">A93</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c13">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_6_1_2"><span class="lab"><font color="Green" title="E55">A100</font></span></a>, <a class="txt" href="#E2:106_1_1_6_1_2"><span class="lab"><font color="Green" title="E56">A101</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>
;<br/><font color="Maroon" title="c10">w</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c8">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_6_1_2_1_1"><span class="lab"><font color="Green" title="E57">A102</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/>.= 
<font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font>

<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c12">e2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E6:106_1_1_6"><span class="lab"><font color="Green" title="E52">A92</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c14">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:106_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E8:106_1_1_6"><span class="lab"><font color="Green" title="E53">A93</font></span></a>, <a class="txt" href="#E1:106_1_1_6_1_2"><span class="lab"><font color="Green" title="E55">A100</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c14">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_6_1_2"><span class="lab"><font color="Green" title="E55">A100</font></span></a>, <a class="txt" href="#E2:106_1_1_6_1_2"><span class="lab"><font color="Green" title="E56">A101</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:106_1_1_6_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1_6"><span class="lab"><font color="Green" title="E50">A90</font></span></a>, <a class="txt" href="#E9:106_1_1_6"><span class="lab"><font color="Green" title="E54">A94</font></span></a>, <a class="txt" href="#E2:106_1_1_6_1_2_1_1"><span class="lab"><font color="Green" title="E58">A103</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_6_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:106_1_1_6_1_2_1_2"/><span class="lab"><font color="Green" title="E57">A104</font></span>: 
<font color="Maroon" title="c12">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c8">H</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"> contradiction</span><br/><div class="add"><span class="lab"><font color="Green" title="E58">A105</font></span>: <font color="Maroon" title="c9">v</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c8">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_6_1_2_1_2"><span class="lab"><font color="Green" title="E57">A104</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/>.= 
<font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font>

<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c12">e2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E6:106_1_1_6"><span class="lab"><font color="Green" title="E52">A92</font></span></a></span>
<br/>.= 
 the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c14">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:106_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E8:106_1_1_6"><span class="lab"><font color="Green" title="E53">A93</font></span></a>, <a class="txt" href="#E1:106_1_1_6_1_2"><span class="lab"><font color="Green" title="E55">A100</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c13">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_6_1_2"><span class="lab"><font color="Green" title="E55">A100</font></span></a>, <a class="txt" href="#E2:106_1_1_6_1_2"><span class="lab"><font color="Green" title="E56">A101</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>
;<br/><font color="Maroon" title="c10">w</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c8">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_6_1_2_1_2"><span class="lab"><font color="Green" title="E57">A104</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/>.= 
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font>

<br/>.= 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c13">a</font>,<font color="Maroon" title="c14">b</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E6:106_1_1_6"><span class="lab"><font color="Green" title="E52">A92</font></span></a>, <a class="txt" href="#E8:106_1_1_6"><span class="lab"><font color="Green" title="E53">A93</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c14">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_6_1_2"><span class="lab"><font color="Green" title="E55">A100</font></span></a>, <a class="txt" href="#E2:106_1_1_6_1_2"><span class="lab"><font color="Green" title="E56">A101</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:106_1_1_6_1_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1_6"><span class="lab"><font color="Green" title="E50">A90</font></span></a>, <a class="txt" href="#E9:106_1_1_6"><span class="lab"><font color="Green" title="E54">A94</font></span></a>, <a class="txt" href="#E2:106_1_1_6_1_2_1_2"><span class="lab"><font color="Green" title="E58">A105</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></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div>

<span class="kw">set </span><font color="Maroon" title="c11">e2</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="106_1_1_7"><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">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c1">G</font> ) implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c8">H</font> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:106_1_1_7"/>
 for <font color="Olive" title="b1">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font>,<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"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c8">H</font></span><br/><a NAME="E2:106_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A106</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> :  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c1">G</font> </span> } </span> 
 ;<br/><a NAME="E3:106_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A107</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></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="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c8">H</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_7_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="E1:106_1_1_7_1"/>
( <font color="Maroon" title="c9">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">w</font> or <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">w</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_7_1_1"><span class="kw">suppose </span></a><a NAME="E1:106_1_1_7_1_1"/><span class="lab"><font color="Green" title="E52">A108</font></span>: 
<font color="Maroon" title="c9">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">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"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c8">H</font></span><br/><div class="add"><a NAME="E2:106_1_1_7_1_1"/>
(  the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp;  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_7_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="E1:106_1_1_7_1_1_1"/><span class="kw">then </span>
( ( <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp; <font color="Maroon" title="c10">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) or ( <font color="Maroon" title="c10">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp; <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_7_1_1"><span class="lab"><font color="Green" title="E52">A108</font></span></a>, <a class="ref" href="glibpre0.html#T4" onmouseover="rs('glibpre0/T4')" onmouseout="rh()">GLIBPRE0:4</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_7_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:106_1_1_7_1_1_1_1"/><span class="lab"><font color="Green" title="E53">A109</font></span>: 
( <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp; <font color="Maroon" title="c10">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c8">H</font></span><br/><div class="add"><span class="lab"><font color="Green" title="E54">A110</font></span>: <font color="Maroon" title="c9">v</font> = 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_7_1_1_1_1"><span class="lab"><font color="Green" title="E53">A109</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E2:106_1_1_7"><span class="lab"><font color="Green" title="E50">A106</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c8">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>

;<br/><font color="Maroon" title="c10">w</font> = 
<font color="Maroon">H<sub>2</sub></font>(<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:106_1_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="#E1:106_1_1_7_1_1"><span class="lab"><font color="Green" title="E52">A108</font></span></a>, <a class="txt" href="#E1:106_1_1_7_1_1_1_1"><span class="lab"><font color="Green" title="E53">A109</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E2:106_1_1_7"><span class="lab"><font color="Green" title="E50">A106</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c8">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>

;<br/><span class="kw">hence </span><a NAME="E4:106_1_1_7_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c8">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:106_1_1_7"><span class="lab"><font color="Green" title="E51">A107</font></span></a>, <a class="txt" href="#E2:106_1_1_7_1_1_1_1"><span class="lab"><font color="Green" title="E54">A110</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</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="106_1_1_7_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:106_1_1_7_1_1_1_2"/><span class="lab"><font color="Green" title="E53">A111</font></span>: 
( <font color="Maroon" title="c10">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp; <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"> the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c8">H</font></span><br/><div class="add"><span class="lab"><font color="Green" title="E54">A112</font></span>: <font color="Maroon" title="c10">w</font> = 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_7_1_1_1_2"><span class="lab"><font color="Green" title="E53">A111</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E2:106_1_1_7"><span class="lab"><font color="Green" title="E50">A106</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c8">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>

;<br/><font color="Maroon" title="c9">v</font> = 
<font color="Maroon">H<sub>2</sub></font>(<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:106_1_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="#E1:106_1_1_7_1_1"><span class="lab"><font color="Green" title="E52">A108</font></span></a>, <a class="txt" href="#E1:106_1_1_7_1_1_1_2"><span class="lab"><font color="Green" title="E53">A111</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E2:106_1_1_7"><span class="lab"><font color="Green" title="E50">A106</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c8">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>

;<br/><span class="kw">hence </span><a NAME="E4:106_1_1_7_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c8">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:106_1_1_7"><span class="lab"><font color="Green" title="E51">A107</font></span></a>, <a class="txt" href="#E2:106_1_1_7_1_1_1_2"><span class="lab"><font color="Green" title="E54">A112</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1_7_1_2"><span class="kw">suppose </span></a><a NAME="E1:106_1_1_7_1_2"/><span class="lab"><font color="Green" title="E52">A113</font></span>: 
<font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">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"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c8">H</font></span><br/><div class="add"><a NAME="E2:106_1_1_7_1_2"/><span class="lab"><font color="Green" title="E53">A114</font></span>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c9">v</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T29" onmouseover="rs('enumset1/T29')" onmouseout="rh()">ENUMSET1:29</a></span>;<br/><span class="lab"><font color="Green" title="E54">A115</font></span>: <font color="Maroon" title="c9">v</font> = 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_7_1_2"><span class="lab"><font color="Green" title="E52">A113</font></span></a>, <a class="txt" href="#E2:106_1_1_7_1_2"><span class="lab"><font color="Green" title="E53">A114</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:106_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E2:106_1_1_7"><span class="lab"><font color="Green" title="E50">A106</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c8">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>

;<br/><font color="Maroon" title="c10">w</font> = 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p3"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:106_1_1_7_1_2"><span class="lab"><font color="Green" title="E52">A113</font></span></a>, <a class="txt" href="#E2:106_1_1_7_1_2"><span class="lab"><font color="Green" title="E53">A114</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:106_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E1:106_1_1_7_1_2"><span class="lab"><font color="Green" title="E52">A113</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:106_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E2:106_1_1_7"><span class="lab"><font color="Green" title="E50">A106</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c8">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>

;<br/><span class="kw">hence </span><a NAME="E5:106_1_1_7_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>,<span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c8">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:106_1_1_7"><span class="lab"><font color="Green" title="E51">A107</font></span></a>, <a class="txt" href="#E3:106_1_1_7_1_2"><span class="lab"><font color="Green" title="E54">A115</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</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="E19:106_1_1"/>
( (  for <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c8">H</font> ) implies  ex <font color="Olive" title="b1">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font>,<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"> verum</span><br/>


</div>
