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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">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <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">G1</font> being    <a href="glib_007.html#M4" title="GLIB_007:mode.4">addAdjVertexAll</a> of <font color="Maroon" title="c1">G2</font>,<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">V</font><br/>  for <font color="Olive" title="b4">v1</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b3">G1</font><br/>  for <font color="Olive" title="b5">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G2</font>  st <font color="Olive" title="b4">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">v2</font> holds <br/>( <font color="Olive" title="b4">v1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">v2</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp; <font color="Olive" title="b4">v1</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">v2</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp; <font color="Olive" title="b4">v1</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">v2</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">v</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">  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">G1</font> being    <a href="glib_007.html#M4" title="GLIB_007:mode.4">addAdjVertexAll</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">v</font>,<font color="Olive" title="b1">V</font><br/>  for <font color="Olive" title="b3">v1</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b2">G1</font><br/>  for <font color="Olive" title="b4">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G2</font>  st <font color="Olive" title="b3">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">v2</font> holds <br/>( <font color="Olive" title="b3">v1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">v2</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp; <font color="Olive" title="b3">v1</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">v2</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp; <font color="Olive" title="b3">v1</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">v2</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">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">G1</font> being    <a href="glib_007.html#M4" title="GLIB_007:mode.4">addAdjVertexAll</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">V</font><br/>  for <font color="Olive" title="b2">v1</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b1">G1</font><br/>  for <font color="Olive" title="b3">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G2</font>  st <font color="Olive" title="b2">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v2</font> holds <br/>( <font color="Olive" title="b2">v1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">v2</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp; <font color="Olive" title="b2">v1</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">v2</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp; <font color="Olive" title="b2">v1</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">v2</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">G1</font> be    <a href="glib_007.html#M4" title="GLIB_007:mode.4">addAdjVertexAll</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">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">v1</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c4">G1</font><br/>  for <font color="Olive" title="b2">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G2</font>  st <font color="Olive" title="b1">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v2</font> holds <br/>( <font color="Olive" title="b1">v1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v2</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp; <font color="Olive" title="b1">v1</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v2</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp; <font color="Olive" title="b1">v1</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v2</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">v1</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c4">G1</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">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G2</font>  st <font color="Maroon" title="c5">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v2</font> holds <br/>( <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v2</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp; <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v2</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp; <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v2</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">v2</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <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="c5">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v2</font> implies ( <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp; <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp; <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 ) )</span><br/>











<span class="kw">assume </span><a NAME="E1:64"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
<font color="Maroon" title="c5">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v2</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="c5">v1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp; <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp; <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_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="E1:64_1"/>
( ( <font color="Maroon" title="c3">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</font> &amp;  not <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</font> &amp; <font color="Maroon" title="c6">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> ) or  not <font color="Maroon" title="c6">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> or  not <font color="Maroon" title="c3">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1"><span class="kw">suppose </span></a><a NAME="E1:64_1_1"/><span class="lab"><font color="Green" title="E11">A2</font></span>: 
( <font color="Maroon" title="c3">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</font> &amp;  not <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</font> &amp; <font color="Maroon" title="c6">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">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="c5">v1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp; <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp; <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">E</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:64_1_1"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c3">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c7">E</font> &amp; <font color="Maroon" title="c7">E</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font> )
 <span class="kw">and </span><br/><a NAME="E4:64_1_1"/>
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c7">E</font>
 <span class="kw">and </span><br/><a NAME="E5:64_1_1"/><span class="lab"><font color="Green" title="E13">A4</font></span>: 
 for <font color="Olive" title="b1">v3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> holds <br/> ex <font color="Olive" title="b2">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">E</font> &amp; <font color="Olive" title="b2">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v3</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> &amp; (  for <font color="Olive" title="b3">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b3">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v3</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> holds <br/><font color="Olive" title="b2">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">e2</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_007.html#D4" onmouseover="rs('glib_007/D4')" onmouseout="rh()">GLIB_007:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c8">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E7:64_1_1"/><span class="lab"><font color="Green" title="E14">A5</font></span>: 
( <font color="Maroon" title="c8">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">E</font> &amp; <font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">v2</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> )
 <span class="kw">and </span><br/><a NAME="E8:64_1_1"/><span class="lab"><font color="Green" title="E15">A6</font></span>: 
 for <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">v2</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> holds <br/><font color="Maroon" title="c8">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E5:64_1_1"><span class="lab"><font color="Green" title="E13">A4</font></span></a></span>;<br/><a NAME="E9:64_1_1"/><span class="lab"><font color="Green" title="E16">A7</font></span>: 
<font color="Maroon" title="c6">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1"><span class="lab"><font color="Green" title="E11">A2</font></span></a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="64_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  holds <br/><font color="Olive" title="b1">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">e1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">e2</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">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">e1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/><span class="kw">assume </span><a NAME="E1:64_1_1_1"/>
<font color="Maroon" title="c9">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</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="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">e1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:64_1_1_1"/><span class="lab"><font color="Green" title="E17">A8</font></span>: 
<font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c4">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T57" onmouseover="rs('glib_000/T57')" onmouseout="rh()">GLIB_000:57</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:64_1_1_1_1"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> or <font color="Maroon" title="c10">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:64_1_1_1_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">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="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">e1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:64_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_1_1"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/><a NAME="E3:64_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">e1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E8:64_1_1"><span class="lab"><font color="Green" title="E15">A6</font></span></a></span>;<br/><a NAME="E4:64_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">e1</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="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:64_1_1_1_1_1"/>
<font color="Maroon" title="c9">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">e1</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#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</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="64_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:64_1_1_1_1_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">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="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">e1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:64_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A9</font></span>: 
<font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c1">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E1:64_1_1"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E9:64_1_1"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="txt" href="#E3:64_1_1_1"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <a class="ref" href="glib_007.html#D4" onmouseover="rs('glib_007/D4')" onmouseout="rh()">GLIB_007:def 4</a></span>;<br/><a NAME="E3:64_1_1_1_1_2"/>
<font color="Maroon" title="c9">e2</font> is    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" onmouseover="rs('tarski/T1')" onmouseout="rh()">TARSKI:1</a></span>;<br/><a NAME="E4:64_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E2:64_1_1_1_1_2"><span class="lab"><font color="Green" title="E18">A9</font></span></a>, <a class="ref" href="glib_000.html#T57" onmouseover="rs('glib_000/T57')" onmouseout="rh()">GLIB_000:57</a></span>;<br/><span class="kw">hence </span><a NAME="E5:64_1_1_1_1_2"/>
<font color="Maroon" title="c9">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">e1</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#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</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="E11:64_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A10</font></span>: 
<font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">e1</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="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="64_1_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  holds <br/><font color="Olive" title="b1">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">e1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">e2</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">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">e1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/><span class="kw">assume </span><a NAME="E1:64_1_1_2"/>
<font color="Maroon" title="c9">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</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="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">e1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:64_1_1_2"/><span class="lab"><font color="Green" title="E18">A11</font></span>: 
<font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c4">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T59" onmouseover="rs('glib_000/T59')" onmouseout="rh()">GLIB_000:59</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_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="E1:64_1_1_2_1"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> or <font color="Maroon" title="c10">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:64_1_1_2_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">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="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">e1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:64_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_1_2"><span class="lab"><font color="Green" title="E18">A11</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/><a NAME="E3:64_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">e1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E8:64_1_1"><span class="lab"><font color="Green" title="E15">A6</font></span></a></span>;<br/><a NAME="E4:64_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">e1</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="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:64_1_1_2_1_1"/>
<font color="Maroon" title="c9">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">e1</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#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</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="64_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:64_1_1_2_1_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">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="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">e1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:64_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A12</font></span>: 
<font color="Maroon" title="c9">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c1">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E1:64_1_1"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E9:64_1_1"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="txt" href="#E3:64_1_1_2"><span class="lab"><font color="Green" title="E18">A11</font></span></a>, <a class="ref" href="glib_007.html#D4" onmouseover="rs('glib_007/D4')" onmouseout="rh()">GLIB_007:def 4</a></span>;<br/><a NAME="E3:64_1_1_2_1_2"/>
<font color="Maroon" title="c9">e2</font> is    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" onmouseover="rs('tarski/T1')" onmouseout="rh()">TARSKI:1</a></span>;<br/><a NAME="E4:64_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E2:64_1_1_2_1_2"><span class="lab"><font color="Green" title="E19">A12</font></span></a>, <a class="ref" href="glib_000.html#T59" onmouseover="rs('glib_000/T59')" onmouseout="rh()">GLIB_000:59</a></span>;<br/><span class="kw">hence </span><a NAME="E5:64_1_1_2_1_2"/>
<font color="Maroon" title="c9">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">e1</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#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</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="E13:64_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A13</font></span>: 
<font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">e1</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="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/><a NAME="E14:64_1_1"/><span class="lab"><font color="Green" title="E19">A14</font></span>: 
( <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  or <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="64_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:64_1_1_3"/><span class="lab"><font color="Green" title="E20">A15</font></span>: 
( <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  &amp; <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:64_1_1_3"/>
<font color="Maroon" title="c1">G2</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c4">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_006.html#T57" onmouseover="rs('glib_006/T57')" onmouseout="rh()">GLIB_006:57</a></span>;<br/>
<a NAME="E3:64_1_1_3"/><span class="kw">then </span>
( <font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  &amp; <font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="glib_000.html#T78" onmouseover="rs('glib_000/T78')" onmouseout="rh()">GLIB_000:78</a></span>;<br/>
<a NAME="E4:64_1_1_3"/><span class="kw">then </span>
(  not <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  &amp;  not <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1_1_3"><span class="lab"><font color="Green" title="E20">A15</font></span></a>, <a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E5:64_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A16</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<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></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E7:64_1_1_3"/><span class="lab"><font color="Green" title="E22">A17</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E9:64_1_1_3"/><span class="lab"><font color="Green" title="E23">A18</font></span>: 
<font color="Maroon" title="c10">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:64_1_1_3"><span class="lab"><font color="Green" title="E21">A16</font></span></a>, <a class="ref" href="xboole_0.html#D1" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E10:64_1_1_3"/><span class="lab"><font color="Green" title="E24">A19</font></span>: 
( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  &amp;  not <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  &amp; <font color="Maroon" title="c10">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  &amp;  not <font color="Maroon" title="c10">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:64_1_1_3"><span class="lab"><font color="Green" title="E22">A17</font></span></a>, <a class="txt" href="#E9:64_1_1_3"><span class="lab"><font color="Green" title="E23">A18</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="E11:64_1_1_3"/><span class="kw">then </span>
( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">e1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c10">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">e1</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="#E11:64_1_1"><span class="lab"><font color="Green" title="E17">A10</font></span></a>, <a class="txt" href="#E13:64_1_1"><span class="lab"><font color="Green" title="E18">A13</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E12:64_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A20</font></span>: 
( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">e1</font> &amp; <font color="Maroon" title="c10">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">e1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">j</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E14:64_1_1_3"/><span class="lab"><font color="Green" title="E26">A21</font></span>: 
<font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c4">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:64_1_1_3"><span class="lab"><font color="Green" title="E24">A19</font></span></a>, <a class="ref" href="glib_000.html#T57" onmouseover="rs('glib_000/T57')" onmouseout="rh()">GLIB_000:57</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">p</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E16:64_1_1_3"/><span class="lab"><font color="Green" title="E27">A22</font></span>: 
<font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c12">p</font>,<font color="Maroon" title="c4">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:64_1_1_3"><span class="lab"><font color="Green" title="E24">A19</font></span></a>, <a class="txt" href="#E12:64_1_1_3"><span class="lab"><font color="Green" title="E25">A20</font></span></a>, <a class="ref" href="glib_000.html#T59" onmouseover="rs('glib_000/T59')" onmouseout="rh()">GLIB_000:59</a></span>;<br/>
<a NAME="E17:64_1_1_3"/>
( <font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v1</font> &amp; <font color="Maroon" title="c5">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:64_1_1_3"><span class="lab"><font color="Green" title="E26">A21</font></span></a>, <a class="txt" href="#E16:64_1_1_3"><span class="lab"><font color="Green" title="E27">A22</font></span></a>, <a class="ref" href="glib_000.html#T125" onmouseover="rs('glib_000/T125')" onmouseout="rh()">GLIB_000:125</a></span>;<br/>
<a NAME="E18:64_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c8">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c5">v1</font>,<font color="Maroon" title="c4">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:64_1_1_3"><span class="lab"><font color="Green" title="E26">A21</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:64_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E7:64_1_1"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="txt" href="#E9:64_1_1"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="ref" href="glib_000.html#T15" onmouseover="rs('glib_000/T15')" onmouseout="rh()">GLIB_000:15</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E15:64_1_1"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">e1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64_1_1"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="txt" href="#E7:64_1_1"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a>, <a class="ref" href="xboole_1.html#T63" onmouseover="rs('xboole_1/T63')" onmouseout="rh()">XBOOLE_1:63</a></span>;<br/><a NAME="E16:64_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A23</font></span>: 
( <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">e1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">e1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T63" onmouseover="rs('xboole_1/T63')" onmouseout="rh()">XBOOLE_1:63</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A24</font></span>:  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">e1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">e1</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="card_2.html#T35" onmouseover="rs('card_2/T35')" onmouseout="rh()">CARD_2:35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T30" onmouseover="rs('card_1/T30')" onmouseout="rh()">CARD_1:30</a></span>
;<br/><span class="lab"><font color="Green" title="E22">A25</font></span>:  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">e1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">e1</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="#E16:64_1_1"><span class="lab"><font color="Green" title="E20">A23</font></span></a>, <a class="ref" href="card_2.html#T35" onmouseover="rs('card_2/T35')" onmouseout="rh()">CARD_2:35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T30" onmouseover="rs('card_1/T30')" onmouseout="rh()">CARD_1:30</a></span>
;<br/><span class="kw">thus </span><a NAME="E19:64_1_1"/>
<font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1
  <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="c5">v1</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp; <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="64_1_1_6"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_6_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:64_1_1_6_1"/>
( <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  or <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:64_1_1"><span class="lab"><font color="Green" title="E19">A14</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_6_1_1"><span class="kw">suppose </span></a><a NAME="E1:64_1_1_6_1_1"/>
<font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</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="c5">v1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1</span><br/><div class="add"><a NAME="E2:64_1_1_6_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A26</font></span>: 
<font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a> 
 ;<br/><a NAME="E3:64_1_1_6_1_1"/>
<font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:64_1_1"><span class="lab"><font color="Green" title="E18">A13</font></span></a>, <a class="txt" href="#E17:64_1_1"><span class="lab"><font color="Green" title="E21">A24</font></span></a>, <a class="ref" href="card_1.html#T11" onmouseover="rs('card_1/T11')" onmouseout="rh()">CARD_1:11</a></span>;<br/><a NAME="E4:64_1_1_6_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_1_1_6_1_1"><span class="lab"><font color="Green" title="E23">A26</font></span></a>, <a class="ref" href="card_2.html#T84" onmouseover="rs('card_2/T84')" onmouseout="rh()">CARD_2:84</a></span>;<br/><span class="kw">hence </span><a NAME="E5:64_1_1_6_1_1"/>
<font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T19" onmouseover="rs('card_2/T19')" onmouseout="rh()">CARD_2:19</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_6_1_2"><span class="kw">suppose </span></a><a NAME="E1:64_1_1_6_1_2"/>
<font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</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="c5">v1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1</span><br/><div class="add"><a NAME="E2:64_1_1_6_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A27</font></span>: 
<font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a> 
 ;<br/><a NAME="E3:64_1_1_6_1_2"/>
<font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:64_1_1"><span class="lab"><font color="Green" title="E17">A10</font></span></a>, <a class="txt" href="#E18:64_1_1"><span class="lab"><font color="Green" title="E22">A25</font></span></a>, <a class="ref" href="card_1.html#T11" onmouseover="rs('card_1/T11')" onmouseout="rh()">CARD_1:11</a></span>;<br/><a NAME="E4:64_1_1_6_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64_1_1_6_1_2"><span class="lab"><font color="Green" title="E23">A27</font></span></a>, <a class="ref" href="card_2.html#T84" onmouseover="rs('card_2/T84')" onmouseout="rh()">CARD_2:84</a></span>;<br/><span class="kw">hence </span><a NAME="E5:64_1_1_6_1_2"/>
<font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T19" onmouseover="rs('card_2/T19')" onmouseout="rh()">CARD_2:19</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E20:64_1_1"/>
( <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp; <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:64_1_1"><span class="lab"><font color="Green" title="E17">A10</font></span></a>, <a class="txt" href="#E13:64_1_1"><span class="lab"><font color="Green" title="E18">A13</font></span></a>, <a class="txt" href="#E17:64_1_1"><span class="lab"><font color="Green" title="E21">A24</font></span></a>, <a class="txt" href="#E18:64_1_1"><span class="lab"><font color="Green" title="E22">A25</font></span></a>, <a class="ref" href="card_1.html#T11" onmouseover="rs('card_1/T11')" onmouseout="rh()">CARD_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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_2"><span class="kw">suppose </span></a><a NAME="E1:64_1_2"/>
 not <font color="Maroon" title="c6">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">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="c5">v1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp; <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp; <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 )</span><br/><div class="add"><a NAME="E2:64_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  &amp; <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  &amp; <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="glibpre0.html#T56" target="_self" onmouseover="rs('glibpre0/T56')" onmouseout="rh()">Th62</a></span>;<br/><span class="kw">hence </span><a NAME="E3:64_1_2"/>
( <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp; <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp; <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T94" onmouseover="rs('card_2/T94')" onmouseout="rh()">CARD_2:94</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="64_1_3"><span class="kw">suppose </span></a><a NAME="E1:64_1_3"/>
(  not <font color="Maroon" title="c3">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">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="c5">v1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp; <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp; <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 )</span><br/><div class="add"><a NAME="E2:64_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c4">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#D4" onmouseover="rs('glib_007/D4')" onmouseout="rh()">GLIB_007:def 4</a></span>;<br/><a NAME="E3:64_1_3"/><span class="kw">then </span>
( <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  &amp; <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  &amp; <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="glib_000.html#T96" onmouseover="rs('glib_000/T96')" onmouseout="rh()">GLIB_000:96</a></span>;<br/><span class="kw">hence </span><a NAME="E4:64_1_3"/>
( <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp; <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 &amp; <font color="Maroon" title="c5">v1</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v2</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T94" onmouseover="rs('card_2/T94')" onmouseout="rh()">CARD_2:94</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>

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