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<span class="kw">let </span><font color="Maroon" title="c1">G2</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">E</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">G1</font> being    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c1">G2</font>,<font color="Olive" title="b1">E</font><br/>  for <font color="Olive" title="b3">v</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b4">G3</font> being   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Olive" title="b2">G1</font>,<font color="Olive" title="b3">v</font><br/>  for <font color="Olive" title="b5">G4</font> being   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G2</font>,<font color="Olive" title="b3">v</font> holds  <font color="Olive" title="b5">G4</font> is    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Olive" title="b4">G3</font>,<font color="Olive" title="b1">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b3">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">E</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G1</font> being    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">E</font><br/>  for <font color="Olive" title="b2">v</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b3">G3</font> being   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Olive" title="b1">G1</font>,<font color="Olive" title="b2">v</font><br/>  for <font color="Olive" title="b4">G4</font> being   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G2</font>,<font color="Olive" title="b2">v</font> holds  <font color="Olive" title="b4">G4</font> is    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Olive" title="b3">G3</font>,<font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">G1</font> be    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">E</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">v</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">G3</font> being   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G1</font>,<font color="Olive" title="b1">v</font><br/>  for <font color="Olive" title="b3">G4</font> being   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G2</font>,<font color="Olive" title="b1">v</font> holds  <font color="Olive" title="b3">G4</font> is    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Olive" title="b2">G3</font>,<font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">v</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G3</font> being   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G1</font>,<font color="Maroon" title="c4">v</font><br/>  for <font color="Olive" title="b2">G4</font> being   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c4">v</font> holds  <font color="Olive" title="b2">G4</font> is    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Olive" title="b1">G3</font>,<font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">G3</font> be   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G1</font>,<font color="Maroon" title="c4">v</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">G4</font> being   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c4">v</font> holds  <font color="Olive" title="b1">G4</font> is    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c5">G3</font>,<font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">G4</font> be   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c4">v</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="c6">G4</font> is    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c5">G3</font>,<font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><br/>











<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_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="E1:29_1"/>
( <font color="Maroon" title="c2">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font> or  not <font color="Maroon" title="c2">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1"><span class="kw">suppose </span></a><a NAME="E1:29_1_1"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
<font color="Maroon" title="c2">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</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="c6">G4</font> is    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c5">G3</font>,<font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><br/><div class="add"><a NAME="E2:29_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A2</font></span>: 
(  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c3">G1</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">G2</font> &amp;  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c3">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font> &amp;  <a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c3">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c1">G2</font></span>)</span> <a href="funct_4.html#K9" title="FUNCT_4:func.9">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">E</font></span>)</span> &amp;  <a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c3">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c1">G2</font></span>)</span> <a href="funct_4.html#K9" title="FUNCT_4:func.9">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">E</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_007.html#D1" target="_self" onmouseover="rs('glib_007/D1')" onmouseout="rh()">Def1</a></span>;<br/><a NAME="E3:29_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A4</font></span>: 
<font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <span class="p1">(<span class="default"><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="c3">G1</font></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" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G2</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <span class="p1">(<span class="default"><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">G2</font></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" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_007.html#T11" target="_self" onmouseover="rs('glib_007/T11')" onmouseout="rh()">Th11</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_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="E1:29_1_1_1"/>
(  not <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</font> or <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:29_1_1_1_1"/><span class="lab"><font color="Green" title="E29">A5</font></span>: 
 not <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</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="c6">G4</font> is    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c5">G3</font>,<font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><br/><div class="add"><a NAME="E2:29_1_1_1_1"/><span class="kw">then </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="c3">G1</font></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" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <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="c3">G1</font> &amp; <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">G2</font></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" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <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">G2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="zfmisc_1.html#T57" onmouseover="rs('zfmisc_1/T57')" onmouseout="rh()">ZFMISC_1:57</a></span>;<br/><a NAME="E3:29_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A6</font></span>: 
( <font color="Maroon" title="c3">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c5">G3</font> &amp; <font color="Maroon" title="c1">G2</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c6">G4</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T94" onmouseover="rs('glib_000/T94')" onmouseout="rh()">GLIB_000:94</a></span>;<br/><a NAME="E4:29_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</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="c3">G1</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> 
 <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>, <a class="txt" href="#E2:29_1_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E1:29_1_1_1_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="ref" href="zfmisc_1.html#T50" onmouseover="rs('zfmisc_1/T50')" onmouseout="rh()">ZFMISC_1:50</a></span>;<br/><a NAME="E5:29_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">a7</font></span>: 
<font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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="glib_000.html#T103" onmouseover="rs('glib_000/T103')" onmouseout="rh()">GLIB_000:103</a></span>;<br/><a NAME="E6:29_1_1_1_1"/>
<font color="Maroon" title="c5">G3</font> is    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_1_1_1_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="ref" href="glib_007.html#T2" target="_self" onmouseover="rs('glib_007/T2')" onmouseout="rh()">Th2</a></span>;<br/><a NAME="E7:29_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">G2</font> is    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c5">G3</font>,<font color="Maroon" title="c2">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_007.html#T3" target="_self" onmouseover="rs('glib_007/T3')" onmouseout="rh()">Th3</a></span>;<br/><span class="kw">hence </span><a NAME="E8:29_1_1_1_1"/>
<font color="Maroon" title="c6">G4</font> is    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c5">G3</font>,<font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_1_1_1_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E5:29_1_1_1_1"><span class="lab"><font color="Green" title="E31">a7</font></span></a>, <a class="ref" href="glib_007.html#T2" target="_self" onmouseover="rs('glib_007/T2')" onmouseout="rh()">Th2</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="29_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:29_1_1_1_2"/><span class="lab"><font color="Green" title="E29">A8</font></span>: 
<font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</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="c6">G4</font> is    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c5">G3</font>,<font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><br/><div class="add"><a NAME="E2:29_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A9</font></span>: 
<font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c3">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_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="E1:29_1_1_1_2_1"/>
(  not <font color="Maroon" title="c1">G2</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  or <font color="Maroon" title="c1">G2</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:29_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E31">A10</font></span>: 
 not <font color="Maroon" title="c1">G2</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">G4</font> is    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c5">G3</font>,<font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><br/><div class="add"><a NAME="E2:29_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A11</font></span>: 
(  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c6">G4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">G2</font></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" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c6">G4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G2</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <span class="p1">(<span class="default"><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">G2</font></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" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1_1_1_2"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="ref" href="glib_000.html#T47" onmouseover="rs('glib_000/T47')" onmouseout="rh()">GLIB_000:47</a></span>;<br/><a NAME="E3:29_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E33">A12</font></span>: 
(  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c5">G3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">G1</font></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" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c5">G3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <span class="p1">(<span class="default"><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="c3">G1</font></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" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E1:29_1_1_1_2"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="ref" href="glib_000.html#T47" onmouseover="rs('glib_000/T47')" onmouseout="rh()">GLIB_000:47</a>, <a class="txt" href="#E1:29_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E31">A10</font></span></a></span>;<br/><a NAME="E4:29_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A13</font></span>: 
(  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c6">G4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c5">G3</font> &amp;  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c6">G4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c5">G3</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="txt" href="#E2:29_1_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E3:29_1_1"><span class="lab"><font color="Green" title="E28">A4</font></span></a></span>;<br/><a NAME="E5:29_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E35">A14</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <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="c6">G4</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c6">G4</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <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="c6">G4</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c6">G4</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T4" onmouseover="rs('glib_000/T4')" onmouseout="rh()">GLIB_000:4</a></span>;<br/><a NAME="E6:29_1_1_1_2_1_1"/>
 for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds <br/><font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c5">G3</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">e</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="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> implies <font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c5">G3</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:29_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span><br/>

<a NAME="E2:29_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A15</font></span>: 
( <font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font> &amp;  not <font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E3:29_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A16</font></span>: 
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c3">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1_1"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E2:29_1_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a></span>;<br/>
<a NAME="E4:29_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E38">A17</font></span>: 
(  not <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="c3">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</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"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  not <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="c3">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</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"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E36">A15</font></span></a>, <a class="txt" href="#E1:29_1_1"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E2:29_1_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="glib_000.html#T28" onmouseover="rs('glib_000/T28')" onmouseout="rh()">GLIB_000:28</a></span>;<br/>
<a NAME="E5:29_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c7">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <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="c3">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</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="c3">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font>,<font color="Maroon" title="c3">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1_1"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E2:29_1_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E2:29_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E36">A15</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="E6:29_1_1_1_2_1_1_1"/><span class="kw">then </span>
( <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="c3">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c3">G1</font> &amp; <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="c3">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c3">G1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T13" onmouseover="rs('glib_000/T13')" onmouseout="rh()">GLIB_000:13</a></span>;<br/>
<a NAME="E7:29_1_1_1_2_1_1_1"/><span class="kw">then </span>
( <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="c3">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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="c3">G1</font></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" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <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="c3">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</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="c3">G1</font></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" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E38">A17</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:29_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c5">G3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="txt" href="#E3:29_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E37">A16</font></span></a>, <a class="ref" href="glib_000.html#T31" onmouseover="rs('glib_000/T31')" onmouseout="rh()">GLIB_000:31</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E7:29_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A18</font></span>: 
<font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c5">G3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/><a NAME="E8:29_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E37">A19</font></span>: 
 for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c6">G4</font> holds <br/> not <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_2_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">e</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="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c6">G4</font> implies  not <font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:29_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c6">G4</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">  not <font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<a NAME="E2:29_1_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A20</font></span>: 
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c5">G3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E34">A13</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:29_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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"> contradiction</span><br/>

<a NAME="E4:29_1_1_1_2_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c3">G1</font> &amp; ( <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="c3">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</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"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <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="c3">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</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"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T28" onmouseover="rs('glib_000/T28')" onmouseout="rh()">GLIB_000:28</a></span>;<br/>
<a NAME="E5:29_1_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A21</font></span>: 
( <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="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</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"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <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="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</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"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E38">A20</font></span></a>, <a class="ref" href="glib_000.html#D32" onmouseover="rs('glib_000/D32')" onmouseout="rh()">GLIB_000:def 32</a></span>;<br/>
<a NAME="E6:29_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c7">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <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="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</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="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font>,<font color="Maroon" title="c5">G3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E38">A20</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="E7:29_1_1_1_2_1_1_2"/><span class="kw">then </span>
( <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="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c5">G3</font> &amp; <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="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c5">G3</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T13" onmouseover="rs('glib_000/T13')" onmouseout="rh()">GLIB_000:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:29_1_1_1_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E39">A21</font></span></a>, <a class="txt" href="#E3:29_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</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">set </span><font color="Maroon" title="c7">S</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="c5">G3</font></span>)</span> <a href="funct_4.html#K9" title="FUNCT_4:func.9">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>;<br/><span class="lab"><font color="Green" title="E38">A22</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T4" onmouseover="rs('glib_000/T4')" onmouseout="rh()">GLIB_000:4</a></span>
;<br/><span class="lab"><font color="Green" title="E39">A23</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_4.html#K9" title="FUNCT_4:func.9">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T4" onmouseover="rs('glib_000/T4')" onmouseout="rh()">GLIB_000:4</a></span>
<br/>.= 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <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="c6">G4</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="txt" href="#E5:29_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E35">A14</font></span></a>, <a class="txt" href="#E9:29_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E38">A22</font></span></a>, <a class="ref" href="xboole_1.html#T22" onmouseover="rs('xboole_1/T22')" onmouseout="rh()">XBOOLE_1:22</a></span>
;<br/><span class="kw">set </span><font color="Maroon" title="c8">T</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="c5">G3</font></span>)</span> <a href="funct_4.html#K9" title="FUNCT_4:func.9">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>;<br/><span class="lab"><font color="Green" title="E40">A24</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T4" onmouseover="rs('glib_000/T4')" onmouseout="rh()">GLIB_000:4</a></span>
;<br/><span class="lab"><font color="Green" title="E41">A25</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_4.html#K9" title="FUNCT_4:func.9">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T4" onmouseover="rs('glib_000/T4')" onmouseout="rh()">GLIB_000:4</a></span>
<br/>.= 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <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="c6">G4</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="txt" href="#E5:29_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E35">A14</font></span></a>, <a class="txt" href="#E11:29_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E40">A24</font></span></a>, <a class="ref" href="xboole_1.html#T22" onmouseover="rs('xboole_1/T22')" onmouseout="rh()">XBOOLE_1:22</a></span>
;<br/><a NAME="E13:29_1_1_1_2_1_1"/>
 for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <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="c6">G4</font></span>)</span> holds <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="c6">G4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_4.html#K9" title="FUNCT_4:func.9">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_2_1_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">e</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="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <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="c6">G4</font></span>)</span> implies <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="c6">G4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_4.html#K9" title="FUNCT_4:func.9">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:29_1_1_1_2_1_1_7"/>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <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="c6">G4</font></span>)</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">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c6">G4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_4.html#K9" title="FUNCT_4:func.9">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font></span><br/>

<a NAME="E2:29_1_1_1_2_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A26</font></span>: 
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c6">G4</font>
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1_2_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="E1:29_1_1_1_2_1_1_7_1"/>
( <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> or  not <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1_2_1_1_7_1_1"><span class="kw">suppose </span></a><a NAME="E1:29_1_1_1_2_1_1_7_1_1"/><span class="lab"><font color="Green" title="E43">A27</font></span>: 
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</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">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c6">G4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_4.html#K9" title="FUNCT_4:func.9">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font></span><br/><div class="add"><a NAME="E2:29_1_1_1_2_1_1_7_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A28</font></span>: 
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font>
 ;<br/><a NAME="E3:29_1_1_1_2_1_1_7_1_1"/><span class="lab"><font color="Green" title="E45">A29</font></span>: 
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c5">G3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1_1_1_2_1_1_7_1_1"><span class="lab"><font color="Green" title="E43">A27</font></span></a>, <a class="txt" href="#E7:29_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E36">A18</font></span></a></span>;<br/><a NAME="E4:29_1_1_1_2_1_1_7_1_1"/><span class="lab"><font color="Green" title="E46">A30</font></span>: 
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1_1_1_2_1_1_7_1_1"><span class="lab"><font color="Green" title="E43">A27</font></span></a>, <a class="txt" href="#E7:29_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E36">A18</font></span></a>, <a class="txt" href="#E9:29_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E38">A22</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="E5:29_1_1_1_2_1_1_7_1_1"/>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_1_1_2_1_1_7"><span class="lab"><font color="Green" title="E42">A26</font></span></a></span>;<br/><a NAME="E6:29_1_1_1_2_1_1_7_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <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="c1">G2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T4" onmouseover="rs('glib_000/T4')" onmouseout="rh()">GLIB_000:4</a></span>;<br/><a NAME="E7:29_1_1_1_2_1_1_7_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G2</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c2">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_1_1_2_1_1_7_1_1"><span class="lab"><font color="Green" title="E44">A28</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="E8:29_1_1_1_2_1_1_7_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A31</font></span>: 
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">E</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_4.html#K9" title="FUNCT_4:func.9">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_1_1_1_2_1_1_7_1_1"><span class="lab"><font color="Green" title="E46">A30</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</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="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1_1_1_2_1_1_7_1_1"><span class="lab"><font color="Green" title="E43">A27</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</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="c3">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_1_1_1_2_1_1_7_1_1"><span class="lab"><font color="Green" title="E45">A29</font></span></a>, <a class="ref" href="glib_000.html#D32" onmouseover="rs('glib_000/D32')" onmouseout="rh()">GLIB_000:def 32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">E</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E8:29_1_1_1_2_1_1_7_1_1"><span class="lab"><font color="Green" title="E47">A31</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</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="c1">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_1_1_2_1_1_7_1_1"><span class="lab"><font color="Green" title="E44">A28</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</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="c6">G4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_1_1_2_1_1_7"><span class="lab"><font color="Green" title="E42">A26</font></span></a>, <a class="ref" href="glib_000.html#D32" onmouseover="rs('glib_000/D32')" onmouseout="rh()">GLIB_000:def 32</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="29_1_1_1_2_1_1_7_1_2"><span class="kw">suppose </span></a><a NAME="E1:29_1_1_1_2_1_1_7_1_2"/><span class="lab"><font color="Green" title="E43">A32</font></span>: 
 not <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</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">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c6">G4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_4.html#K9" title="FUNCT_4:func.9">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font></span><br/><div class="add"><a NAME="E2:29_1_1_1_2_1_1_7_1_2"/>
 not <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_1_1_2_1_1_7"><span class="lab"><font color="Green" title="E42">A26</font></span></a>, <a class="txt" href="#E8:29_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E37">A19</font></span></a></span>;<br/><a NAME="E3:29_1_1_1_2_1_1_7_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A33</font></span>: 
 not <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">E</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1_1_1_2_1_1_7_1_2"><span class="lab"><font color="Green" title="E43">A32</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E4:29_1_1_1_2_1_1_7_1_2"/><span class="lab"><font color="Green" title="E45">A34</font></span>: 
 not <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1_1_1_2_1_1_7_1_2"><span class="lab"><font color="Green" title="E43">A32</font></span></a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_4.html#K9" title="FUNCT_4:func.9">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</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="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_1_1_1_2_1_1_7_1_2"><span class="lab"><font color="Green" title="E45">A34</font></span></a>, <a class="ref" href="funct_4.html#T11" onmouseover="rs('funct_4/T11')" onmouseout="rh()">FUNCT_4:11</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="c3">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="txt" href="#E2:29_1_1_1_2_1_1_7"><span class="lab"><font color="Green" title="E42">A26</font></span></a>, <a class="ref" href="glib_000.html#D32" onmouseover="rs('glib_000/D32')" onmouseout="rh()">GLIB_000:def 32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E3:29_1_1_1_2_1_1_7_1_2"><span class="lab"><font color="Green" title="E44">A33</font></span></a>, <a class="ref" href="funct_4.html#T11" onmouseover="rs('funct_4/T11')" onmouseout="rh()">FUNCT_4:11</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="c6">G4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_1_1_2_1_1_7"><span class="lab"><font color="Green" title="E42">A26</font></span></a>, <a class="ref" href="glib_000.html#D32" onmouseover="rs('glib_000/D32')" onmouseout="rh()">GLIB_000:def 32</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><a NAME="E14:29_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A35</font></span>: 
 <a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c6">G4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c5">G3</font></span>)</span> <a href="funct_4.html#K9" title="FUNCT_4:func.9">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:29_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E39">A23</font></span></a>, <a class="ref" href="funct_1.html#T2" onmouseover="rs('funct_1/T2')" onmouseout="rh()">FUNCT_1:2</a></span>;<br/><a NAME="E15:29_1_1_1_2_1_1"/>
 for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <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="c6">G4</font></span>)</span> holds <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="c6">G4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_4.html#K9" title="FUNCT_4:func.9">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_2_1_1_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">e</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="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <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="c6">G4</font></span>)</span> implies <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="c6">G4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_4.html#K9" title="FUNCT_4:func.9">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:29_1_1_1_2_1_1_8"/>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <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="c6">G4</font></span>)</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">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c6">G4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_4.html#K9" title="FUNCT_4:func.9">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font></span><br/>

<a NAME="E2:29_1_1_1_2_1_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A36</font></span>: 
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c6">G4</font>
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1_2_1_1_8_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="E1:29_1_1_1_2_1_1_8_1"/>
( <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> or  not <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1_2_1_1_8_1_1"><span class="kw">suppose </span></a><a NAME="E1:29_1_1_1_2_1_1_8_1_1"/><span class="lab"><font color="Green" title="E44">A37</font></span>: 
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</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">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c6">G4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_4.html#K9" title="FUNCT_4:func.9">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font></span><br/><div class="add"><a NAME="E2:29_1_1_1_2_1_1_8_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A38</font></span>: 
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font>
 ;<br/><a NAME="E3:29_1_1_1_2_1_1_8_1_1"/><span class="lab"><font color="Green" title="E46">A39</font></span>: 
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c5">G3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1_1_1_2_1_1_8_1_1"><span class="lab"><font color="Green" title="E44">A37</font></span></a>, <a class="txt" href="#E7:29_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E36">A18</font></span></a></span>;<br/><a NAME="E4:29_1_1_1_2_1_1_8_1_1"/><span class="lab"><font color="Green" title="E47">A40</font></span>: 
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1_1_1_2_1_1_8_1_1"><span class="lab"><font color="Green" title="E44">A37</font></span></a>, <a class="txt" href="#E7:29_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E36">A18</font></span></a>, <a class="txt" href="#E11:29_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E40">A24</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="E5:29_1_1_1_2_1_1_8_1_1"/>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_1_1_2_1_1_8"><span class="lab"><font color="Green" title="E43">A36</font></span></a></span>;<br/><a NAME="E6:29_1_1_1_2_1_1_8_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <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="c1">G2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T4" onmouseover="rs('glib_000/T4')" onmouseout="rh()">GLIB_000:4</a></span>;<br/><a NAME="E7:29_1_1_1_2_1_1_8_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G2</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c2">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_1_1_2_1_1_8_1_1"><span class="lab"><font color="Green" title="E45">A38</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="E8:29_1_1_1_2_1_1_8_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A41</font></span>: 
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">E</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_4.html#K9" title="FUNCT_4:func.9">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_1_1_1_2_1_1_8_1_1"><span class="lab"><font color="Green" title="E47">A40</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</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="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1_1_1_2_1_1_8_1_1"><span class="lab"><font color="Green" title="E44">A37</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</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="c3">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_1_1_1_2_1_1_8_1_1"><span class="lab"><font color="Green" title="E46">A39</font></span></a>, <a class="ref" href="glib_000.html#D32" onmouseover="rs('glib_000/D32')" onmouseout="rh()">GLIB_000:def 32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">E</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E8:29_1_1_1_2_1_1_8_1_1"><span class="lab"><font color="Green" title="E48">A41</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</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="c1">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_1_1_2_1_1_8_1_1"><span class="lab"><font color="Green" title="E45">A38</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</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="c6">G4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_1_1_2_1_1_8"><span class="lab"><font color="Green" title="E43">A36</font></span></a>, <a class="ref" href="glib_000.html#D32" onmouseover="rs('glib_000/D32')" onmouseout="rh()">GLIB_000:def 32</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="29_1_1_1_2_1_1_8_1_2"><span class="kw">suppose </span></a><a NAME="E1:29_1_1_1_2_1_1_8_1_2"/><span class="lab"><font color="Green" title="E44">A42</font></span>: 
 not <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</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">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c6">G4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_4.html#K9" title="FUNCT_4:func.9">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font></span><br/><div class="add"><a NAME="E2:29_1_1_1_2_1_1_8_1_2"/>
 not <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_1_1_2_1_1_8"><span class="lab"><font color="Green" title="E43">A36</font></span></a>, <a class="txt" href="#E8:29_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E37">A19</font></span></a></span>;<br/><a NAME="E3:29_1_1_1_2_1_1_8_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A43</font></span>: 
 not <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">E</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1_1_1_2_1_1_8_1_2"><span class="lab"><font color="Green" title="E44">A42</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E4:29_1_1_1_2_1_1_8_1_2"/>
 not <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1_1_1_2_1_1_8_1_2"><span class="lab"><font color="Green" title="E44">A42</font></span></a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="funct_4.html#K9" title="FUNCT_4:func.9">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</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="c5">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T11" onmouseover="rs('funct_4/T11')" onmouseout="rh()">FUNCT_4:11</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="c3">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="txt" href="#E2:29_1_1_1_2_1_1_8"><span class="lab"><font color="Green" title="E43">A36</font></span></a>, <a class="ref" href="glib_000.html#D32" onmouseover="rs('glib_000/D32')" onmouseout="rh()">GLIB_000:def 32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E3:29_1_1_1_2_1_1_8_1_2"><span class="lab"><font color="Green" title="E45">A43</font></span></a>, <a class="ref" href="funct_4.html#T11" onmouseover="rs('funct_4/T11')" onmouseout="rh()">FUNCT_4:11</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="c6">G4</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_1_1_2_1_1_8"><span class="lab"><font color="Green" title="E43">A36</font></span></a>, <a class="ref" href="glib_000.html#D32" onmouseover="rs('glib_000/D32')" onmouseout="rh()">GLIB_000:def 32</a></span>
;<br/> <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><a NAME="E16:29_1_1_1_2_1_1"/><span class="kw">then </span>
 <a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c6">G4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c5">G3</font></span>)</span> <a href="funct_4.html#K9" title="FUNCT_4:func.9">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G3</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:29_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E41">A25</font></span></a>, <a class="ref" href="funct_1.html#T2" onmouseover="rs('funct_1/T2')" onmouseout="rh()">FUNCT_1:2</a></span>;<br/><span class="kw">hence </span><a NAME="E17:29_1_1_1_2_1_1"/>
<font color="Maroon" title="c6">G4</font> is    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c5">G3</font>,<font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="txt" href="#E7:29_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E36">A18</font></span></a>, <a class="txt" href="#E14:29_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E42">A35</font></span></a>, <a class="ref" href="glib_007.html#D1" target="_self" onmouseover="rs('glib_007/D1')" onmouseout="rh()">Def1</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="29_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:29_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E31">A45</font></span>: 
<font color="Maroon" title="c1">G2</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">G4</font> is    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c5">G3</font>,<font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">v1</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G1</font><span class="kw"> such that </span><br/><a NAME="E3:29_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E32">A47</font></span>: 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c3">G1</font> <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="c7">v1</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="glib_000.html#T22" onmouseover="rs('glib_000/T22')" onmouseout="rh()">GLIB_000:22</a></span>;<br/><a NAME="E4:29_1_1_1_2_1_2"/>
<font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_1_1_2"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="txt" href="#E3:29_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E32">A47</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E5:29_1_1_1_2_1_2"/><span class="kw">then </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="c3">G1</font></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" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; <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">G2</font></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" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T37" onmouseover="rs('xboole_1/T37')" onmouseout="rh()">XBOOLE_1:37</a>, <a class="txt" href="#E2:29_1_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E3:29_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E32">A47</font></span></a></span>;<br/><a NAME="E6:29_1_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A48</font></span>: 
( <font color="Maroon" title="c3">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c5">G3</font> &amp; <font color="Maroon" title="c1">G2</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c6">G4</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D37" onmouseover="rs('glib_000/D37')" onmouseout="rh()">GLIB_000:def 37</a></span>;<br/><a NAME="E7:29_1_1_1_2_1_2"/>
<font color="Maroon" title="c5">G3</font> is    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:29_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E33">A48</font></span></a>, <a class="ref" href="glib_007.html#T2" target="_self" onmouseover="rs('glib_007/T2')" onmouseout="rh()">Th2</a></span>;<br/><a NAME="E8:29_1_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">G3</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c1">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E31">A45</font></span></a>, <a class="ref" href="glib_007.html#T6" target="_self" onmouseover="rs('glib_007/T6')" onmouseout="rh()">Th6</a></span>;<br/><span class="kw">hence </span><a NAME="E9:29_1_1_1_2_1_2"/>
<font color="Maroon" title="c6">G4</font> is    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c5">G3</font>,<font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_007.html#T6" target="_self" onmouseover="rs('glib_007/T6')" onmouseout="rh()">Th6</a>, <a class="txt" href="#E1:29_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E31">A45</font></span></a>, <a class="txt" href="#E6:29_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E33">A48</font></span></a>, <a class="ref" href="glib_000.html#T85" onmouseover="rs('glib_000/T85')" onmouseout="rh()">GLIB_000:85</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2"><span class="kw">suppose </span></a><a NAME="E1:29_1_2"/><span class="lab"><font color="Green" title="E26">A50</font></span>: 
 not <font color="Maroon" title="c2">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</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="c6">G4</font> is    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c5">G3</font>,<font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><br/><div class="add"><a NAME="E2:29_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A51</font></span>: 
<font color="Maroon" title="c3">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c1">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_007.html#D1" target="_self" onmouseover="rs('glib_007/D1')" onmouseout="rh()">Def1</a></span>;<br/><a NAME="E3:29_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A52</font></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="c3">G1</font></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" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">G2</font></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" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D34" onmouseover="rs('glib_000/D34')" onmouseout="rh()">GLIB_000:def 34</a></span>;<br/><a NAME="E4:29_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A53</font></span>: 
<font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <span class="p1">(<span class="default"><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="c3">G1</font></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" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G2</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <span class="p1">(<span class="default"><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">G2</font></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" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_2"><span class="lab"><font color="Green" title="E27">A51</font></span></a>, <a class="ref" href="glib_000.html#T90" onmouseover="rs('glib_000/T90')" onmouseout="rh()">GLIB_000:90</a></span>;<br/><a NAME="E5:29_1_2"/><span class="lab"><font color="Green" title="E30">A54</font></span>: 
 not <font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c5">G3</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:29_1_2_1"/>
<font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c5">G3</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="E2:29_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c3">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/>
<a NAME="E3:29_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c3">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/>
<a NAME="E4:29_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A55</font></span>: 
<font color="Maroon" title="c2">E</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c3">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T39" onmouseover="rs('xboole_1/T39')" onmouseout="rh()">XBOOLE_1:39</a></span>;<br/>
<a NAME="E5:29_1_2_1"/>
 not <font color="Maroon" title="c2">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c3">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1_2"><span class="lab"><font color="Green" title="E26">A50</font></span></a>, <a class="txt" href="#E2:29_1_2"><span class="lab"><font color="Green" title="E27">A51</font></span></a>, <a class="ref" href="glib_000.html#D34" onmouseover="rs('glib_000/D34')" onmouseout="rh()">GLIB_000:def 34</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:29_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_1_2_1"><span class="lab"><font color="Green" title="E31">A55</font></span></a>, <a class="ref" href="xboole_1.html#T11" onmouseover="rs('xboole_1/T11')" onmouseout="rh()">XBOOLE_1:11</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="E6:29_1_2"/>
<font color="Maroon" title="c6">G4</font> is   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">G1</font>,<font color="Maroon" title="c4">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_2"><span class="lab"><font color="Green" title="E27">A51</font></span></a>, <a class="txt" href="#E3:29_1_2"><span class="lab"><font color="Green" title="E28">A52</font></span></a>, <a class="txt" href="#E4:29_1_2"><span class="lab"><font color="Green" title="E29">A53</font></span></a>, <a class="ref" href="glib_000.html#T95" onmouseover="rs('glib_000/T95')" onmouseout="rh()">GLIB_000:95</a></span>;<br/><a NAME="E7:29_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">G3</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c6">G4</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T93" onmouseover="rs('glib_000/T93')" onmouseout="rh()">GLIB_000:93</a></span>;<br/><span class="kw">hence </span><a NAME="E8:29_1_2"/>
<font color="Maroon" title="c6">G4</font> is    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c5">G3</font>,<font color="Maroon" title="c2">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesInOut</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29_1_2"><span class="lab"><font color="Green" title="E30">A54</font></span></a>, <a class="ref" href="glib_007.html#D1" target="_self" onmouseover="rs('glib_007/D1')" onmouseout="rh()">Def1</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>
