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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>, <font color="Olive" title="b2">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <br/>  for <font color="Olive" title="b3">w</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G2</font><br/>  for <font color="Olive" title="b4">G1</font> being    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c1">G2</font>,<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">e</font>,<font color="Olive" title="b3">w</font><br/>  for <font color="Olive" title="b5">w1</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b4">G1</font>  st  not <font color="Olive" title="b2">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> &amp;  not <font color="Olive" title="b1">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="Olive" title="b5">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">w</font> holds <br/>( <font color="Olive" title="b5">w1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">w</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="Olive" title="b2">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b5">w1</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">w</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="b5">w1</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="Olive" title="b3">w</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  &amp; <font color="Olive" title="b5">w1</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="Olive" title="b3">w</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  &amp; <font color="Olive" title="b5">w1</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">w</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</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="Olive" title="b2">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b5">w1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">w</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/><span class="kw">let </span><font color="Maroon" title="c2">v</font>, <font color="Maroon" title="c3">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">  for <font color="Olive" title="b1">w</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G2</font><br/>  for <font color="Olive" title="b2">G1</font> being    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">e</font>,<font color="Olive" title="b1">w</font><br/>  for <font color="Olive" title="b3">w1</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b2">G1</font>  st  not <font color="Maroon" title="c3">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> &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="Olive" title="b3">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">w</font> holds <br/>( <font color="Olive" title="b3">w1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">w</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="c3">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b3">w1</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">w</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">w1</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="Olive" title="b1">w</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  &amp; <font color="Olive" title="b3">w1</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="Olive" title="b1">w</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  &amp; <font color="Olive" title="b3">w1</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">w</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</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="c3">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b3">w1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">w</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/><span class="kw">let </span><font color="Maroon" title="c4">w</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">  for <font color="Olive" title="b1">G1</font> being    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">e</font>,<font color="Maroon" title="c4">w</font><br/>  for <font color="Olive" title="b2">w1</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b1">G1</font>  st  not <font color="Maroon" title="c3">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> &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="Olive" title="b2">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w</font> holds <br/>( <font color="Olive" title="b2">w1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</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="c3">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b2">w1</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</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">w1</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="c4">w</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  &amp; <font color="Olive" title="b2">w1</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="c4">w</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  &amp; <font color="Olive" title="b2">w1</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</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="c3">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b2">w1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</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/><span class="kw">let </span><font color="Maroon" title="c5">G1</font> be    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">e</font>,<font color="Maroon" title="c4">w</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">w1</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c5">G1</font>  st  not <font color="Maroon" title="c3">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> &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="Olive" title="b1">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w</font> holds <br/>( <font color="Olive" title="b1">w1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</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="c3">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b1">w1</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</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">w1</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="c4">w</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  &amp; <font color="Olive" title="b1">w1</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="c4">w</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  &amp; <font color="Olive" title="b1">w1</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</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="c3">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b1">w1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</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/><span class="kw">let </span><font color="Maroon" title="c6">w1</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c5">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"> (  not <font color="Maroon" title="c3">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> &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">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w</font> implies ( <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</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="c3">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</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="c6">w1</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="c4">w</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  &amp; <font color="Maroon" title="c6">w1</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="c4">w</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  &amp; <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</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="c3">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</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/>









<span class="kw">assume </span><a NAME="E1:10"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
(  not <font color="Maroon" title="c3">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> &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">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</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="c3">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</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="c6">w1</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="c4">w</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  &amp; <font color="Maroon" title="c6">w1</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="c4">w</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  &amp; <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</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="c3">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</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/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">G3</font> being   <a href="glib_006.html#NM3" title="GLIB_006:NM.3">addVertex</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">v</font><span class="kw"> such that </span><br/><a NAME="E3:10"/><span class="lab"><font color="Green" title="E11">A2</font></span>: 
<font color="Maroon" title="c5">G1</font> is    <a href="glib_006.html#M3" title="GLIB_006:mode.3">addEdge</a> of <font color="Maroon" title="c7">G3</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">e</font>,<font color="Maroon" title="c4">w</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_006.html#T126" onmouseover="rs('glib_006/T126')" onmouseout="rh()">GLIB_006:126</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">w3</font> = <font color="Maroon" title="c4">w</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c7">G3</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_006.html#T68" onmouseover="rs('glib_006/T68')" onmouseout="rh()">GLIB_006:68</a></span>;<br/>
<a NAME="E5:10"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
( <font color="Maroon" title="c8">w3</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="c4">w</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  &amp; <font color="Maroon" title="c8">w3</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="c4">w</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  &amp; <font color="Maroon" title="c8">w3</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="c4">w</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  &amp; <font color="Maroon" title="c8">w3</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="c4">w</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  &amp; <font color="Maroon" title="c8">w3</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  &amp; <font color="Maroon" title="c8">w3</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="c4">w</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glibpre0.html#T45" onmouseover="rs('glibpre0/T45')" onmouseout="rh()">GLIBPRE0:45</a></span>;<br/>
<a NAME="E6:10"/><span class="lab"><font color="Green" title="E13">A4</font></span>: 
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c7">G3</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_006.html#D10" onmouseover="rs('glib_006/D10')" onmouseout="rh()">GLIB_006:def 10</a></span>;<br/>
<a NAME="E7:10"/>
<font color="Maroon" title="c2">v</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="c2">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="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E8:10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A5</font></span>: 
<font color="Maroon" title="c2">v</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c7">G3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_006.html#T86" onmouseover="rs('glib_006/T86')" onmouseout="rh()">GLIB_006:86</a></span>;<br/>
<a NAME="E9:10"/>
<font color="Maroon" title="c2">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">w</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E10">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:10"/>
( <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K26" title="GLIB_000:func.26">.edgesIn()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</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="c3">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K30" title="GLIB_000:func.30">.inDegree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</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="c6">w1</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="c4">w</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesOut()</a>  &amp; <font color="Maroon" title="c6">w1</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="c4">w</font> <a href="glib_000.html#K31" title="GLIB_000:func.31">.outDegree()</a>  &amp; <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</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="c3">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c6">w1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">w</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="txt" href="#E1:10"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E3:10"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E5:10"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="txt" href="#E6:10"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E8:10"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="ref" href="glibpre0.html#T48" onmouseover="rs('glibpre0/T48')" onmouseout="rh()">GLIBPRE0:48</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>
