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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">G</font> being   <a href="glib_000.html#V2" title="GLIB_000:attr.2">_finite</a>   <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a><br/>  for <font color="Olive" title="b2">H</font> being   <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Olive" title="b1">G</font>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">G</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">H</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> holds <br/> ex <font color="Olive" title="b3">p</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="glib_000.html#V12" title="GLIB_000:attr.12">Graph-yielding</a>   <a href="glib_000.html#V14" title="GLIB_000:attr.14">_finite</a>   <a href="glpacy00.html#V2" title="GLPACY00:attr.2">Path-like</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Olive" title="b2">H</font> &amp; <font color="Olive" title="b3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">G</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">H</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; (  for <font color="Olive" title="b4">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b3">p</font>  st <font color="Olive" title="b4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b5">v1</font>, <font color="Olive" title="b6">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b1">G</font> ex <font color="Olive" title="b7">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Olive" title="b3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">n</font>,<font color="Olive" title="b5">v1</font>,<font color="Olive" title="b7">e</font>,<font color="Olive" title="b6">v2</font> &amp; <font color="Olive" title="b7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Olive" title="b1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Olive" title="b5">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">n</font></span>)</span> &amp;  not <font color="Olive" title="b6">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">n</font></span>)</span> &amp; (  not <font color="Olive" title="b3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b5">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">n</font></span>)</span> ) ) or (  not <font color="Olive" title="b5">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">n</font></span>)</span> &amp; <font color="Olive" title="b6">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">n</font></span>)</span> &amp; (  not <font color="Olive" title="b3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b6">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">n</font></span>)</span> ) ) ) ) ) );<br/>
<a NAME="E1:45"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#V2" title="GLIB_000:attr.2">_finite</a>   <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a>  <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">H</font> being   <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</font>  st  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">H</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> holds <br/> ex <font color="Olive" title="b2">p</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="glib_000.html#V12" title="GLIB_000:attr.12">Graph-yielding</a>   <a href="glib_000.html#V14" title="GLIB_000:attr.14">_finite</a>   <a href="glpacy00.html#V2" title="GLPACY00:attr.2">Path-like</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Olive" title="b1">H</font> &amp; <font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">H</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; (  for <font color="Olive" title="b3">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b2">p</font>  st <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b4">v1</font>, <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">G</font> ex <font color="Olive" title="b6">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font>,<font color="Olive" title="b4">v1</font>,<font color="Olive" title="b6">e</font>,<font color="Olive" title="b5">v2</font> &amp; <font color="Olive" title="b6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Olive" title="b4">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> &amp;  not <font color="Olive" title="b5">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> &amp; (  not <font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b4">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> ) ) or (  not <font color="Olive" title="b4">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> &amp; <font color="Olive" title="b5">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> &amp; (  not <font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b5">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> ) ) ) ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">H</font> be   <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> implies  ex <font color="Olive" title="b1">p</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="glib_000.html#V12" title="GLIB_000:attr.12">Graph-yielding</a>   <a href="glib_000.html#V14" title="GLIB_000:attr.14">_finite</a>   <a href="glpacy00.html#V2" title="GLPACY00:attr.2">Path-like</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">H</font> &amp; <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; (  for <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">p</font>  st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b3">v1</font>, <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">G</font> ex <font color="Olive" title="b5">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font>,<font color="Olive" title="b3">v1</font>,<font color="Olive" title="b5">e</font>,<font color="Olive" title="b4">v2</font> &amp; <font color="Olive" title="b5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Olive" title="b3">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> &amp;  not <font color="Olive" title="b4">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> &amp; (  not <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b3">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> ) ) or (  not <font color="Olive" title="b3">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> &amp; <font color="Olive" title="b4">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> &amp; (  not <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b4">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> ) ) ) ) ) ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:45_1"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="glib_000.html#V12" title="GLIB_000:attr.12">Graph-yielding</a>   <a href="glib_000.html#V14" title="GLIB_000:attr.14">_finite</a>   <a href="glpacy00.html#V2" title="GLPACY00:attr.2">Path-like</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">H</font> &amp; <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; (  for <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">p</font>  st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b3">v1</font>, <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">G</font> ex <font color="Olive" title="b5">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font>,<font color="Olive" title="b3">v1</font>,<font color="Olive" title="b5">e</font>,<font color="Olive" title="b4">v2</font> &amp; <font color="Olive" title="b5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Olive" title="b3">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> &amp;  not <font color="Olive" title="b4">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> &amp; (  not <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b3">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> ) ) or (  not <font color="Olive" title="b3">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> &amp; <font color="Olive" title="b4">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> &amp; (  not <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b4">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> ) ) ) ) ) )</span><br/>

<a NAME="E2:45_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">H</font> is  <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T117" onmouseover="rs('glib_000/T117')" onmouseout="rh()">GLIB_000:117</a></span>;<br/>
<a NAME="E3:45_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_008.html#T21" onmouseover="rs('glib_008/T21')" onmouseout="rh()">GLIB_008:21</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">p</font> = <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">G</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="glib_000.html#V12" title="GLIB_000:attr.12">Graph-yielding</a>   <a href="glib_000.html#V14" title="GLIB_000:attr.14">_finite</a>   <a href="glpacy00.html#V2" title="GLPACY00:attr.2">Path-like</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">p</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="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">H</font> &amp; <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; (  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">p</font>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b2">v1</font>, <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">G</font> ex <font color="Olive" title="b4">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">v1</font>,<font color="Olive" title="b4">e</font>,<font color="Olive" title="b3">v2</font> &amp; <font color="Olive" title="b4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Olive" title="b2">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp;  not <font color="Olive" title="b3">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b2">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) ) or (  not <font color="Olive" title="b2">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; <font color="Olive" title="b3">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b3">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) ) ) ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E5:45_1"/>
<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; (  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">p</font>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b2">v1</font>, <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">G</font> ex <font color="Olive" title="b4">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">v1</font>,<font color="Olive" title="b4">e</font>,<font color="Olive" title="b3">v2</font> &amp; <font color="Olive" title="b4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Olive" title="b2">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp;  not <font color="Olive" title="b3">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b2">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) ) or (  not <font color="Olive" title="b2">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; <font color="Olive" title="b3">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b3">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) ) ) ) ) )</span><br/>

<span class="kw">thus </span><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> = 
<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T40" onmouseover="rs('finseq_1/T40')" onmouseout="rh()">FINSEQ_1:40</a></span>
<br/>.= 
<font color="Maroon" title="c1">G</font>

; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; (  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">p</font>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b2">v1</font>, <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">G</font> ex <font color="Olive" title="b4">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">v1</font>,<font color="Olive" title="b4">e</font>,<font color="Olive" title="b3">v2</font> &amp; <font color="Olive" title="b4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Olive" title="b2">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp;  not <font color="Olive" title="b3">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b2">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) ) or (  not <font color="Olive" title="b2">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; <font color="Olive" title="b3">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b3">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) ) ) ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E7:45_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="finseq_1.html#T40" onmouseover="rs('finseq_1/T40')" onmouseout="rh()">FINSEQ_1:40</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">p</font>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b2">v1</font>, <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">G</font> ex <font color="Olive" title="b4">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">v1</font>,<font color="Olive" title="b4">e</font>,<font color="Olive" title="b3">v2</font> &amp; <font color="Olive" title="b4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Olive" title="b2">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp;  not <font color="Olive" title="b3">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b2">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) ) or (  not <font color="Olive" title="b2">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; <font color="Olive" title="b3">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b3">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) ) ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">p</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 implies  ex <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> ex <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font>,<font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">e</font>,<font color="Olive" title="b2">v2</font> &amp; <font color="Olive" title="b3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Olive" title="b1">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> &amp;  not <font color="Olive" title="b2">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b1">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> ) ) or (  not <font color="Olive" title="b1">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> &amp; <font color="Olive" title="b2">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b2">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> ) ) ) ) )</span><br/>

<a NAME="E8:45_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E9:45_1"/><span class="kw">then </span>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T40" onmouseover="rs('finseq_1/T40')" onmouseout="rh()">FINSEQ_1:40</a></span>;<br/>
<a NAME="E10:45_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A4</font></span>: 
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E11:45_1"/>
<font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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">  ex <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> ex <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font>,<font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">e</font>,<font color="Olive" title="b2">v2</font> &amp; <font color="Olive" title="b3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Olive" title="b1">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> &amp;  not <font color="Olive" title="b2">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b1">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> ) ) or (  not <font color="Olive" title="b1">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> &amp; <font color="Olive" title="b2">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b2">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> ) ) ) )</span><br/>

<a NAME="E12:45_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T40" onmouseover="rs('finseq_1/T40')" onmouseout="rh()">FINSEQ_1:40</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:45_1"/>
 ex <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> ex <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font>,<font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">e</font>,<font color="Olive" title="b2">v2</font> &amp; <font color="Olive" title="b3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Olive" title="b1">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> &amp;  not <font color="Olive" title="b2">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b1">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> ) ) or (  not <font color="Olive" title="b1">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> &amp; <font color="Olive" title="b2">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b2">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font></span>)</span> ) ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:45_1"><span class="lab"><font color="Green" title="E30">A4</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:45"/><span class="lab"><font color="Green" title="E28">A5</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:45_2"/><span class="lab"><font color="Green" title="E29">A6</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</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">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">G</font> be   <a href="glib_000.html#V2" title="GLIB_000:attr.2">_finite</a>   <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a>  <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">H</font> being   <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c2">G</font>  st <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">H</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> holds <br/> ex <font color="Olive" title="b2">p</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="glib_000.html#V12" title="GLIB_000:attr.12">Graph-yielding</a>   <a href="glib_000.html#V14" title="GLIB_000:attr.14">_finite</a>   <a href="glpacy00.html#V2" title="GLPACY00:attr.2">Path-like</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Olive" title="b1">H</font> &amp; <font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">H</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; (  for <font color="Olive" title="b3">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b2">p</font>  st <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b4">v1</font>, <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="c2">G</font> ex <font color="Olive" title="b6">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font>,<font color="Olive" title="b4">v1</font>,<font color="Olive" title="b6">e</font>,<font color="Olive" title="b5">v2</font> &amp; <font color="Olive" title="b6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Olive" title="b4">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> &amp;  not <font color="Olive" title="b5">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> &amp; (  not <font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b4">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> ) ) or (  not <font color="Olive" title="b4">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> &amp; <font color="Olive" title="b5">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> &amp; (  not <font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b5">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> ) ) ) ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">H</font> be   <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c2">G</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> implies  ex <font color="Olive" title="b1">p</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="glib_000.html#V12" title="GLIB_000:attr.12">Graph-yielding</a>   <a href="glib_000.html#V14" title="GLIB_000:attr.14">_finite</a>   <a href="glpacy00.html#V2" title="GLPACY00:attr.2">Path-like</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c3">H</font> &amp; <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; (  for <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">p</font>  st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b3">v1</font>, <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="c2">G</font> ex <font color="Olive" title="b5">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font>,<font color="Olive" title="b3">v1</font>,<font color="Olive" title="b5">e</font>,<font color="Olive" title="b4">v2</font> &amp; <font color="Olive" title="b5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Olive" title="b3">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> &amp;  not <font color="Olive" title="b4">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> &amp; (  not <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b3">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> ) ) or (  not <font color="Olive" title="b3">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> &amp; <font color="Olive" title="b4">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> &amp; (  not <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b4">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> ) ) ) ) ) ) )</span><br/>



<span class="kw">assume </span><a NAME="E2:45_2"/><span class="lab"><font color="Green" title="E30">A7</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="glib_000.html#V12" title="GLIB_000:attr.12">Graph-yielding</a>   <a href="glib_000.html#V14" title="GLIB_000:attr.14">_finite</a>   <a href="glpacy00.html#V2" title="GLPACY00:attr.2">Path-like</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c3">H</font> &amp; <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; (  for <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">p</font>  st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b3">v1</font>, <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="c2">G</font> ex <font color="Olive" title="b5">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font>,<font color="Olive" title="b3">v1</font>,<font color="Olive" title="b5">e</font>,<font color="Olive" title="b4">v2</font> &amp; <font color="Olive" title="b5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Olive" title="b3">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> &amp;  not <font color="Olive" title="b4">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> &amp; (  not <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b3">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> ) ) or (  not <font color="Olive" title="b3">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> &amp; <font color="Olive" title="b4">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> &amp; (  not <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b4">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> ) ) ) ) ) )</span><br/>

<a NAME="E3:45_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A8</font></span>: 
<font color="Maroon" title="c2">G</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
;<br/>
<a NAME="E4:45_2"/>
<font color="Maroon" title="c2">G</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">H</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_2"><span class="lab"><font color="Green" title="E30">A7</font></span></a></span>;<br/>
<a NAME="E5:45_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A9</font></span>: 
 not <font color="Maroon" title="c3">H</font> is  <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T117" onmouseover="rs('glib_000/T117')" onmouseout="rh()">GLIB_000:117</a></span>;<br/>
<a NAME="E6:45_2"/>
<font color="Maroon" title="c2">G</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_2"><span class="lab"><font color="Green" title="E31">A8</font></span></a></span>;<br/>
<a NAME="E7:45_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A10</font></span>: 
 not <font color="Maroon" title="c2">G</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T26" onmouseover="rs('glib_000/T26')" onmouseout="rh()">GLIB_000:26</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font><span class="kw"> such that </span><br/><a NAME="E9:45_2"/><span class="lab"><font color="Green" title="E34">A11</font></span>: 
( <font color="Maroon" title="c4">v</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  &amp;  not <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c3">H</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45_2"><span class="lab"><font color="Green" title="E32">A9</font></span></a>, <a class="ref" href="glib_008.html#T31" onmouseover="rs('glib_008/T31')" onmouseout="rh()">GLIB_008:31</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">e0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E11:45_2"/><span class="lab"><font color="Green" title="E35">A12</font></span>: 
( <font color="Maroon" title="c4">v</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"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">e0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  not <font color="Maroon" title="c5">e0</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c2">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:45_2"><span class="lab"><font color="Green" title="E34">A11</font></span></a>, <a class="ref" href="glib_000.html#D51" onmouseover="rs('glib_000/D51')" onmouseout="rh()">GLIB_000:def 51</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">G2</font> =  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">v</font>;<br/>
<a NAME="E12:45_2"/><span class="lab"><font color="Green" title="E36">A13</font></span>: 
<span class="p1">(<span class="default"> the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">v</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45_2"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="ref" href="glib_000.html#T48" onmouseover="rs('glib_000/T48')" onmouseout="rh()">GLIB_000:48</a></span>;<br/>
<a NAME="E13:45_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A14</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">v</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_2"><span class="lab"><font color="Green" title="E30">A7</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E38">A15</font></span>:  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">v</font> = 
<font color="Maroon" title="c2">G</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45_2"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="ref" href="glib_000.html#T47" onmouseover="rs('glib_000/T47')" onmouseout="rh()">GLIB_000:47</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesBetween</a> <span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T107" onmouseover="rs('glib_000/T107')" onmouseout="rh()">GLIB_000:107</a></span>
<br/>.= 
<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="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">e0</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="#E11:45_2"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="ref" href="glib_000.html#T34" onmouseover="rs('glib_000/T34')" onmouseout="rh()">GLIB_000:34</a></span>
;<br/>
<a NAME="E15:45_2"/><span class="lab"><font color="Green" title="E39">A16</font></span>: 
<font color="Maroon" title="c3">H</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">v</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_2_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:45_2_2"/>
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c3">H</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:45_2"><span class="lab"><font color="Green" title="E34">A11</font></span></a>, <a class="ref" href="zfmisc_1.html#T50" onmouseover="rs('zfmisc_1/T50')" onmouseout="rh()">ZFMISC_1:50</a></span>;<br/>
<a NAME="E2:45_2_2"/><span class="kw">then </span>
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c3">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T86" onmouseover="rs('xboole_1/T86')" onmouseout="rh()">XBOOLE_1:86</a></span>;<br/>
<a NAME="E3:45_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A17</font></span>: 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c3">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45_2"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="ref" href="glib_000.html#T47" onmouseover="rs('glib_000/T47')" onmouseout="rh()">GLIB_000:47</a></span>;<br/>
<a NAME="E4:45_2_2"/>
 not <font color="Maroon" title="c5">e0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c3">H</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:45_2_2_1"/><span class="lab"><font color="Green" title="E41">A18</font></span>: 
<font color="Maroon" title="c5">e0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c3">H</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:45_2_2_1"/>
<font color="Maroon" title="c5">e0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:45_2"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_2_2_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="E1:45_2_2_1_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font> or <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T61" onmouseover="rs('glib_000/T61')" onmouseout="rh()">GLIB_000:61</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_2_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:45_2_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:45_2_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_2_2_1"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="ref" href="glib_000.html#D32" onmouseover="rs('glib_000/D32')" onmouseout="rh()">GLIB_000:def 32</a></span>;<br/><span class="kw">hence </span><a NAME="E3:45_2_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:45_2"><span class="lab"><font color="Green" title="E34">A11</font></span></a>, <a class="txt" href="#E1:45_2_2_1"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="ref" href="funct_2.html#T5" onmouseover="rs('funct_2/T5')" onmouseout="rh()">FUNCT_2:5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_2_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:45_2_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:45_2_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_2_2_1"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="ref" href="glib_000.html#D32" onmouseover="rs('glib_000/D32')" onmouseout="rh()">GLIB_000:def 32</a></span>;<br/><span class="kw">hence </span><a NAME="E3:45_2_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:45_2"><span class="lab"><font color="Green" title="E34">A11</font></span></a>, <a class="txt" href="#E1:45_2_2_1"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="ref" href="funct_2.html#T5" onmouseover="rs('funct_2/T5')" onmouseout="rh()">FUNCT_2:5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E5:45_2_2"/><span class="kw">then </span>
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c3">H</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">e0</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="zfmisc_1.html#T50" onmouseover="rs('zfmisc_1/T50')" onmouseout="rh()">ZFMISC_1:50</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:45_2_2"/>
<font color="Maroon" title="c3">H</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:45_2"><span class="lab"><font color="Green" title="E38">A15</font></span></a>, <a class="txt" href="#E3:45_2_2"><span class="lab"><font color="Green" title="E40">A17</font></span></a>, <a class="ref" href="glib_000.html#T44" onmouseover="rs('glib_000/T44')" onmouseout="rh()">GLIB_000:44</a>, <a class="ref" href="xboole_1.html#T86" onmouseover="rs('xboole_1/T86')" onmouseout="rh()">XBOOLE_1:86</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="E16:45_2"/><span class="lab"><font color="Green" title="E40">A19</font></span>: 
 the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">v</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45_2"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="txt" href="#E9:45_2"><span class="lab"><font color="Green" title="E34">A11</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">p</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="glib_000.html#V12" title="GLIB_000:attr.12">Graph-yielding</a>   <a href="glib_000.html#V14" title="GLIB_000:attr.14">_finite</a>   <a href="glpacy00.html#V2" title="GLPACY00:attr.2">Path-like</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E18:45_2"/><span class="lab"><font color="Green" title="E41">A20</font></span>: 
( <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c3">H</font> &amp; <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">v</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"> the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">v</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 )
 <span class="kw">and </span><br/><a NAME="E19:45_2"/><span class="lab"><font color="Green" title="E42">A21</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">p</font>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b2">v1</font>, <font color="Olive" title="b3">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">v</font> ex <font color="Olive" title="b4">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">v1</font>,<font color="Olive" title="b4">e</font>,<font color="Olive" title="b3">v2</font> &amp; <font color="Olive" title="b4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Olive" title="b2">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp;  not <font color="Olive" title="b3">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b2">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) ) or (  not <font color="Olive" title="b2">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; <font color="Olive" title="b3">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b3">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_2"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="txt" href="#E13:45_2"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="txt" href="#E15:45_2"><span class="lab"><font color="Green" title="E39">A16</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">q</font> = <font color="Maroon" title="c7">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c2">G</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="glib_000.html#V12" title="GLIB_000:attr.12">Graph-yielding</a>   <a href="glib_000.html#V14" title="GLIB_000:attr.14">_finite</a>   <a href="glpacy00.html#V2" title="GLPACY00:attr.2">Path-like</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">q</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="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c3">H</font> &amp; <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; (  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">q</font>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b2">v1</font>, <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="c2">G</font> ex <font color="Olive" title="b4">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">v1</font>,<font color="Olive" title="b4">e</font>,<font color="Olive" title="b3">v2</font> &amp; <font color="Olive" title="b4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Olive" title="b2">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp;  not <font color="Olive" title="b3">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b2">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) ) or (  not <font color="Olive" title="b2">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; <font color="Olive" title="b3">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b3">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) ) ) ) ) )</span><br/>

<a NAME="E21:45_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T20" onmouseover="rs('finseq_1/T20')" onmouseout="rh()">FINSEQ_1:20</a></span>;<br/>
<a NAME="E22:45_2"/><span class="kw">then </span>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:45_2"/>
<font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c3">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:45_2"><span class="lab"><font color="Green" title="E41">A20</font></span></a>, <a class="ref" href="finseq_1.html#D7" onmouseover="rs('finseq_1/D7')" onmouseout="rh()">FINSEQ_1:def 7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; (  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">q</font>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b2">v1</font>, <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="c2">G</font> ex <font color="Olive" title="b4">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">v1</font>,<font color="Olive" title="b4">e</font>,<font color="Olive" title="b3">v2</font> &amp; <font color="Olive" title="b4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Olive" title="b2">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp;  not <font color="Olive" title="b3">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b2">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) ) or (  not <font color="Olive" title="b2">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; <font color="Olive" title="b3">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b3">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) ) ) ) ) )</span><br/>

<span class="lab"><font color="Green" title="E43">A22</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">q</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c2">G</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T40" onmouseover="rs('finseq_1/T40')" onmouseout="rh()">FINSEQ_1:40</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E25:45_2"/><span class="lab"><font color="Green" title="E44">A23</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T42" onmouseover="rs('finseq_1/T42')" onmouseout="rh()">FINSEQ_1:42</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"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; (  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">q</font>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b2">v1</font>, <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="c2">G</font> ex <font color="Olive" title="b4">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">v1</font>,<font color="Olive" title="b4">e</font>,<font color="Olive" title="b3">v2</font> &amp; <font color="Olive" title="b4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Olive" title="b2">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp;  not <font color="Olive" title="b3">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b2">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) ) or (  not <font color="Olive" title="b2">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; <font color="Olive" title="b3">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b3">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) ) ) ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E26:45_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:45_2"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="txt" href="#E18:45_2"><span class="lab"><font color="Green" title="E41">A20</font></span></a>, <a class="txt" href="#E24:45_2"><span class="lab"><font color="Green" title="E43">A22</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">q</font>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b2">v1</font>, <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="c2">G</font> ex <font color="Olive" title="b4">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">v1</font>,<font color="Olive" title="b4">e</font>,<font color="Olive" title="b3">v2</font> &amp; <font color="Olive" title="b4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Olive" title="b2">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp;  not <font color="Olive" title="b3">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b2">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) ) or (  not <font color="Olive" title="b2">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; <font color="Olive" title="b3">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b3">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> ) ) ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c9">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">q</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 implies  ex <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font> ex <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font>,<font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">e</font>,<font color="Olive" title="b2">v2</font> &amp; <font color="Olive" title="b3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Olive" title="b1">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp;  not <font color="Olive" title="b2">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b1">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) or (  not <font color="Olive" title="b1">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; <font color="Olive" title="b2">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b2">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E27:45_2"/>
<font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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">  ex <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font> ex <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font>,<font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">e</font>,<font color="Olive" title="b2">v2</font> &amp; <font color="Olive" title="b3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Olive" title="b1">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp;  not <font color="Olive" title="b2">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b1">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) or (  not <font color="Olive" title="b1">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; <font color="Olive" title="b2">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b2">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) ) )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_2_4"><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="E1:45_2_4"/><span class="kw">then </span>
( <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 or <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="glpacy00.html#E26"><span class="lab"><font color="Green" title="E25">Lm4</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_2_4_1"><span class="kw">suppose </span></a><a NAME="E1:45_2_4_1"/><span class="lab"><font color="Green" title="E45">A24</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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">  ex <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font> ex <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font>,<font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">e</font>,<font color="Olive" title="b2">v2</font> &amp; <font color="Olive" title="b3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Olive" title="b1">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp;  not <font color="Olive" title="b2">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b1">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) or (  not <font color="Olive" title="b1">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; <font color="Olive" title="b2">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b2">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) ) )</span><br/><div class="add"><a NAME="E2:45_2_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A25</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:45_2"><span class="lab"><font color="Green" title="E44">A23</font></span></a></span>;<br/><a NAME="E3:45_2_4_1"/><span class="lab"><font color="Green" title="E47">A26</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E4:45_2_4_1"/><span class="lab"><font color="Green" title="E48">A27</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:45_2"><span class="lab"><font color="Green" title="E43">A22</font></span></a>, <a class="txt" href="#E1:45_2_4_1"><span class="lab"><font color="Green" title="E45">A24</font></span></a></span>;<br/><a NAME="E5:45_2_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_2_4_1"><span class="lab"><font color="Green" title="E47">A26</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E6:45_2_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A28</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:45_2"><span class="lab"><font color="Green" title="E41">A20</font></span></a>, <a class="txt" href="#E4:45_2_4_1"><span class="lab"><font color="Green" title="E48">A27</font></span></a>, <a class="ref" href="finseq_1.html#D7" onmouseover="rs('finseq_1/D7')" onmouseout="rh()">FINSEQ_1:def 7</a></span>;<br/><a NAME="E7:45_2_4_1"/><span class="lab"><font color="Green" title="E50">A29</font></span>: 
<font color="Maroon" title="c5">e0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:45_2"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E8:45_2_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A30</font></span>: 
<font color="Maroon" title="c5">e0</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="c2">G</font>
 ;<br/><a NAME="E9:45_2_4_1"/>
<font color="Maroon" title="c5">e0</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="c5">e0</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="E10:45_2_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A31</font></span>: 
 not <font color="Maroon" title="c5">e0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:45_2"><span class="lab"><font color="Green" title="E38">A15</font></span></a>, <a class="txt" href="#E6:45_2_4_1"><span class="lab"><font color="Green" title="E49">A28</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="E11:45_2_4_1"/>
<font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E12:45_2_4_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E13:45_2_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A32</font></span>: 
 not <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45_2"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="txt" href="#E6:45_2_4_1"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="ref" href="glib_000.html#T47" onmouseover="rs('glib_000/T47')" onmouseout="rh()">GLIB_000:47</a></span>;<br/><a NAME="E14:45_2_4_1"/>
<font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:45_2"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="txt" href="#E7:45_2_4_1"><span class="lab"><font color="Green" title="E50">A29</font></span></a>, <a class="ref" href="glib_000.html#T67" onmouseover="rs('glib_000/T67')" onmouseout="rh()">GLIB_000:67</a></span>;<br/><a NAME="E15:45_2_4_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E16:45_2_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E17:45_2_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A33</font></span>: 
<font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45_2"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="txt" href="#E6:45_2_4_1"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="ref" href="glib_000.html#T47" onmouseover="rs('glib_000/T47')" onmouseout="rh()">GLIB_000:47</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_2_4_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="E1:45_2_4_1_1"/>
( <font color="Maroon" title="c2">G</font> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font>,<font color="Maroon" title="c5">e0</font>,<font color="Maroon" title="c4">v</font> or <font color="Maroon" title="c2">G</font> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">e0</font>,<font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45_2"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="txt" href="#E11:45_2"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="ref" href="glib_008.html#T38" onmouseover="rs('glib_008/T38')" onmouseout="rh()">GLIB_008:38</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_2_4_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:45_2_4_1_1_1"/><span class="lab"><font color="Green" title="E55">A34</font></span>: 
<font color="Maroon" title="c2">G</font> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font>,<font color="Maroon" title="c5">e0</font>,<font color="Maroon" title="c4">v</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font> ex <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font>,<font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">e</font>,<font color="Olive" title="b2">v2</font> &amp; <font color="Olive" title="b3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Olive" title="b1">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp;  not <font color="Olive" title="b2">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b1">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) or (  not <font color="Olive" title="b1">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; <font color="Olive" title="b2">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b2">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) ) )</span><br/><div class="add"><span class="kw">take </span>
<font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font> ex <font color="Olive" title="b2">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font>,<font color="Olive" title="b2">e</font>,<font color="Olive" title="b1">v2</font> &amp; <font color="Olive" title="b2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp;  not <font color="Olive" title="b1">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) or (  not <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; <font color="Olive" title="b1">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b1">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) ) )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c4">v</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font>,<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c4">v</font> &amp; <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp;  not <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) or (  not <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) ) )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c5">e0</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="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font>,<font color="Maroon" title="c5">e0</font>,<font color="Maroon" title="c4">v</font> &amp; <font color="Maroon" title="c5">e0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp;  not <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) or (  not <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) ) )</span><br/><span class="kw">thus </span><a NAME="E2:45_2_4_1_1_1"/>
<font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font>,<font color="Maroon" title="c5">e0</font>,<font color="Maroon" title="c4">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_2_4_1"><span class="lab"><font color="Green" title="E46">A25</font></span></a>, <a class="txt" href="#E6:45_2_4_1"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="txt" href="#E1:45_2_4_1_1_1"><span class="lab"><font color="Green" title="E55">A34</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">e0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp;  not <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) or (  not <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) ) )</span><br/><a NAME="E3:45_2_4_1_1_1"/>
(  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_2_4_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:45_2_4_1_1_1_1"/>
 not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span><br/>

<a NAME="E2:45_2_4_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A35</font></span>: 
 not  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">v</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45_2_4_1"><span class="lab"><font color="Green" title="E49">A28</font></span></a></span>;<br/>
<a NAME="E3:45_2_4_1_1_1_1"/><span class="lab"><font color="Green" title="E57">A36</font></span>: 
 the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">v</font> is  <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:45_2"><span class="lab"><font color="Green" title="E34">A11</font></span></a>, <a class="ref" href="glpacy00.html#T13" target="_self" onmouseover="rs('glpacy00/T13')" onmouseout="rh()">Th21</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">w</font> = <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">v</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45_2_4_1"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="txt" href="#E17:45_2_4_1"><span class="lab"><font color="Green" title="E54">A33</font></span></a></span>;<br/>
<a NAME="E5:45_2_4_1_1_1_1"/>
<font color="Maroon" title="c10">w</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_2_4_1_1_1_1"><span class="lab"><font color="Green" title="E57">A36</font></span></a>, <a class="ref" href="ordinal1.html#T22" onmouseover="rs('ordinal1/T22')" onmouseout="rh()">ORDINAL1:22</a></span>;<br/>
<a NAME="E6:45_2_4_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A37</font></span>: 
<font color="Maroon" title="c10">w</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T51" onmouseover="rs('card_1/T51')" onmouseout="rh()">CARD_1:51</a></span>;<br/>
<a NAME="E7:45_2_4_1_1_1_1"/><span class="lab"><font color="Green" title="E59">A38</font></span>: 
 not <font color="Maroon" title="c10">w</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_2_4_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:45_2_4_1_1_1_1_1"/><span class="lab"><font color="Green" title="E60">A39</font></span>: 
<font color="Maroon" title="c10">w</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font></span>)</span> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">w</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.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="#E6:45_2_4_1"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="txt" href="#E10:45_2_4_1"><span class="lab"><font color="Green" title="E52">A31</font></span></a>, <a class="txt" href="#E13:45_2_4_1"><span class="lab"><font color="Green" title="E53">A32</font></span></a>, <a class="txt" href="#E1:45_2_4_1_1_1"><span class="lab"><font color="Green" title="E55">A34</font></span></a>, <a class="ref" href="glpacy00.html#T6" target="_self" onmouseover="rs('glpacy00/T6')" onmouseout="rh()">Th10</a></span>
<br/>.= 
2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_2_4_1_1_1_1_1"><span class="lab"><font color="Green" title="E60">A39</font></span></a></span>
;<br/>
<a NAME="E3:45_2_4_1_1_1_1_1"/><span class="kw">then </span>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> 2 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glpacy00.html#D1" target="_self" onmouseover="rs('glpacy00/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<a NAME="E4:45_2_4_1_1_1_1_1"/><span class="kw">then </span>
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T21" onmouseover="rs('ordinal1/T21')" onmouseout="rh()">ORDINAL1:21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:45_2_4_1_1_1_1_1"/>
contradiction
 ; <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="E8:45_2_4_1_1_1_1"/>
 not <font color="Maroon" title="c10">w</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:45_2"><span class="lab"><font color="Green" title="E40">A19</font></span></a>, <a class="txt" href="#E2:45_2_4_1_1_1_1"><span class="lab"><font color="Green" title="E56">A35</font></span></a>, <a class="ref" href="glib_000.html#T157" onmouseover="rs('glib_000/T157')" onmouseout="rh()">GLIB_000:157</a></span>;<br/>
<a NAME="E9:45_2_4_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">w</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45_2_4_1_1_1_1"><span class="lab"><font color="Green" title="E58">A37</font></span></a>, <a class="txt" href="#E7:45_2_4_1_1_1_1"><span class="lab"><font color="Green" title="E59">A38</font></span></a>, <a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:45_2_4_1_1_1_1"/>
<font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45_2_4_1"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="ref" href="glib_006.html#T56" onmouseover="rs('glib_006/T56')" onmouseout="rh()">GLIB_006:56</a>, <a class="ref" href="glib_000.html#T174" onmouseover="rs('glib_000/T174')" onmouseout="rh()">GLIB_000:174</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:45_2_4_1_1_1"/>
( <font color="Maroon" title="c5">e0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp;  not <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) or (  not <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:45_2_4_1"><span class="lab"><font color="Green" title="E51">A30</font></span></a>, <a class="txt" href="#E10:45_2_4_1"><span class="lab"><font color="Green" title="E52">A31</font></span></a>, <a class="txt" href="#E13:45_2_4_1"><span class="lab"><font color="Green" title="E53">A32</font></span></a>, <a class="txt" href="#E17:45_2_4_1"><span class="lab"><font color="Green" title="E54">A33</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_2_4_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:45_2_4_1_1_2"/><span class="lab"><font color="Green" title="E55">A40</font></span>: 
<font color="Maroon" title="c2">G</font> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">e0</font>,<font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">v1</font>, <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="c2">G</font> ex <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font>,<font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">e</font>,<font color="Olive" title="b2">v2</font> &amp; <font color="Olive" title="b3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Olive" title="b1">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp;  not <font color="Olive" title="b2">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b1">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) or (  not <font color="Olive" title="b1">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; <font color="Olive" title="b2">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b2">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) ) )</span><br/><div class="add"><span class="kw">take </span>
<font color="Maroon" title="c4">v</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <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="c2">G</font> ex <font color="Olive" title="b2">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c4">v</font>,<font color="Olive" title="b2">e</font>,<font color="Olive" title="b1">v2</font> &amp; <font color="Olive" title="b2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp;  not <font color="Olive" title="b1">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) or (  not <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; <font color="Olive" title="b1">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b1">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) ) )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c4">v</font>,<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> &amp; <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp;  not <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) or (  not <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) ) )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c5">e0</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="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">e0</font>,<font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> &amp; <font color="Maroon" title="c5">e0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp;  not <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) or (  not <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) ) )</span><br/><span class="kw">thus </span><a NAME="E2:45_2_4_1_1_2"/>
<font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">e0</font>,<font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_2_4_1"><span class="lab"><font color="Green" title="E46">A25</font></span></a>, <a class="txt" href="#E6:45_2_4_1"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="txt" href="#E1:45_2_4_1_1_2"><span class="lab"><font color="Green" title="E55">A40</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">e0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp;  not <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) or (  not <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) ) )</span><br/><a NAME="E3:45_2_4_1_1_2"/>
(  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_2_4_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:45_2_4_1_1_2_1"/>
 not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span><br/>

<a NAME="E2:45_2_4_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A41</font></span>: 
 not  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">v</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45_2_4_1"><span class="lab"><font color="Green" title="E49">A28</font></span></a></span>;<br/>
<a NAME="E3:45_2_4_1_1_2_1"/><span class="lab"><font color="Green" title="E57">A42</font></span>: 
 the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">v</font> is  <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:45_2"><span class="lab"><font color="Green" title="E34">A11</font></span></a>, <a class="ref" href="glpacy00.html#T13" target="_self" onmouseover="rs('glpacy00/T13')" onmouseout="rh()">Th21</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">w</font> = <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">v</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45_2_4_1"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="txt" href="#E17:45_2_4_1"><span class="lab"><font color="Green" title="E54">A33</font></span></a></span>;<br/>
<a NAME="E5:45_2_4_1_1_2_1"/>
<font color="Maroon" title="c10">w</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_2_4_1_1_2_1"><span class="lab"><font color="Green" title="E57">A42</font></span></a>, <a class="ref" href="ordinal1.html#T22" onmouseover="rs('ordinal1/T22')" onmouseout="rh()">ORDINAL1:22</a></span>;<br/>
<a NAME="E6:45_2_4_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A43</font></span>: 
<font color="Maroon" title="c10">w</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,2</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T51" onmouseover="rs('card_1/T51')" onmouseout="rh()">CARD_1:51</a></span>;<br/>
<a NAME="E7:45_2_4_1_1_2_1"/><span class="lab"><font color="Green" title="E59">A44</font></span>: 
 not <font color="Maroon" title="c10">w</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_2_4_1_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:45_2_4_1_1_2_1_1"/><span class="lab"><font color="Green" title="E60">A45</font></span>: 
<font color="Maroon" title="c10">w</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font></span>)</span> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">w</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.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="#E6:45_2_4_1"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="txt" href="#E10:45_2_4_1"><span class="lab"><font color="Green" title="E52">A31</font></span></a>, <a class="txt" href="#E13:45_2_4_1"><span class="lab"><font color="Green" title="E53">A32</font></span></a>, <a class="txt" href="#E1:45_2_4_1_1_2"><span class="lab"><font color="Green" title="E55">A40</font></span></a>, <a class="ref" href="glpacy00.html#T7" target="_self" onmouseover="rs('glpacy00/T7')" onmouseout="rh()">Th11</a></span>
<br/>.= 
2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_2_4_1_1_2_1_1"><span class="lab"><font color="Green" title="E60">A45</font></span></a></span>
;<br/>
<a NAME="E3:45_2_4_1_1_2_1_1"/><span class="kw">then </span>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> 2 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glpacy00.html#D1" target="_self" onmouseover="rs('glpacy00/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<a NAME="E4:45_2_4_1_1_2_1_1"/><span class="kw">then </span>
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T21" onmouseover="rs('ordinal1/T21')" onmouseout="rh()">ORDINAL1:21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:45_2_4_1_1_2_1_1"/>
contradiction
 ; <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="E8:45_2_4_1_1_2_1"/>
 not <font color="Maroon" title="c10">w</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:45_2"><span class="lab"><font color="Green" title="E40">A19</font></span></a>, <a class="txt" href="#E2:45_2_4_1_1_2_1"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="ref" href="glib_000.html#T157" onmouseover="rs('glib_000/T157')" onmouseout="rh()">GLIB_000:157</a></span>;<br/>
<a NAME="E9:45_2_4_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">w</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45_2_4_1_1_2_1"><span class="lab"><font color="Green" title="E58">A43</font></span></a>, <a class="txt" href="#E7:45_2_4_1_1_2_1"><span class="lab"><font color="Green" title="E59">A44</font></span></a>, <a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:45_2_4_1_1_2_1"/>
<font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45_2_4_1"><span class="lab"><font color="Green" title="E49">A28</font></span></a>, <a class="ref" href="glib_006.html#T56" onmouseover="rs('glib_006/T56')" onmouseout="rh()">GLIB_006:56</a>, <a class="ref" href="glib_000.html#T174" onmouseover="rs('glib_000/T174')" onmouseout="rh()">GLIB_000:174</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:45_2_4_1_1_2"/>
( <font color="Maroon" title="c5">e0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp;  not <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) or (  not <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c5">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:45_2_4_1"><span class="lab"><font color="Green" title="E51">A30</font></span></a>, <a class="txt" href="#E10:45_2_4_1"><span class="lab"><font color="Green" title="E52">A31</font></span></a>, <a class="txt" href="#E13:45_2_4_1"><span class="lab"><font color="Green" title="E53">A32</font></span></a>, <a class="txt" href="#E17:45_2_4_1"><span class="lab"><font color="Green" title="E54">A33</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_2_4_2"><span class="kw">suppose </span></a><a NAME="E1:45_2_4_2"/><span class="lab"><font color="Green" title="E45">A46</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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">  ex <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font> ex <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font>,<font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">e</font>,<font color="Olive" title="b2">v2</font> &amp; <font color="Olive" title="b3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Olive" title="b1">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp;  not <font color="Olive" title="b2">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b1">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) or (  not <font color="Olive" title="b1">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; <font color="Olive" title="b2">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b2">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) ) )</span><br/><div class="add"><a NAME="E2:45_2_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A47</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/><a NAME="E3:45_2_4_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">m</font> = <font color="Maroon" title="c9">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">p</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_2_4_2"><span class="lab"><font color="Green" title="E46">A47</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">v1</font>, <font color="Maroon" title="c12">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c13">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:45_2_4_2"/><span class="lab"><font color="Green" title="E47">A48</font></span>: 
( <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font>,<font color="Maroon" title="c11">v1</font>,<font color="Maroon" title="c13">e</font>,<font color="Maroon" title="c12">v2</font> &amp; <font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span></span>)</span> &amp; ( ( <font color="Maroon" title="c11">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span> &amp;  not <font color="Maroon" title="c12">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span> &amp; (  not <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c11">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span> ) ) or (  not <font color="Maroon" title="c11">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span> &amp; <font color="Maroon" title="c12">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span> &amp; (  not <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c12">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span> ) ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:45_2"><span class="lab"><font color="Green" title="E42">A21</font></span></a>, <a class="txt" href="#E1:45_2_4_2"><span class="lab"><font color="Green" title="E45">A46</font></span></a></span>;<br/><a NAME="E7:45_2_4_2"/>
( 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_2_4_2"><span class="lab"><font color="Green" title="E45">A46</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E8:45_2_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E9:45_2_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A49</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D7" onmouseover="rs('finseq_1/D7')" onmouseout="rh()">FINSEQ_1:def 7</a></span>;<br/><a NAME="E10:45_2_4_2"/><span class="lab"><font color="Green" title="E49">A50</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D7" onmouseover="rs('finseq_1/D7')" onmouseout="rh()">FINSEQ_1:def 7</a></span>;<br/><a NAME="E11:45_2_4_2"/>
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">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="c2">G</font>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">v1</font> = <font color="Maroon" title="c11">v1</font>, <font color="Maroon" title="c15">v2</font> = <font color="Maroon" title="c12">v2</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c14">v1</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font> ex <font color="Olive" title="b2">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c14">v1</font>,<font color="Olive" title="b2">e</font>,<font color="Olive" title="b1">v2</font> &amp; <font color="Olive" title="b2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Maroon" title="c14">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp;  not <font color="Olive" title="b1">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c14">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) or (  not <font color="Maroon" title="c14">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; <font color="Olive" title="b1">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b1">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) ) )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c15">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">  ex <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c14">v1</font>,<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c15">v2</font> &amp; <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Maroon" title="c14">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp;  not <font color="Maroon" title="c15">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c14">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) or (  not <font color="Maroon" title="c14">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; <font color="Maroon" title="c15">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c15">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) ) )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c13">e</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c13">e</font>,<font color="Maroon" title="c15">v2</font> &amp; <font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Maroon" title="c14">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp;  not <font color="Maroon" title="c15">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c14">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) or (  not <font color="Maroon" title="c14">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; <font color="Maroon" title="c15">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c15">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) ) )</span><br/><a NAME="E13:45_2_4_2"/>
<span class="p1">(<span class="default"><a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a>  the   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T33" onmouseover="rs('xboole_1/T33')" onmouseout="rh()">XBOOLE_1:33</a></span>;<br/><span class="kw">hence </span><a NAME="E14:45_2_4_2"/>
( <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c13">e</font>,<font color="Maroon" title="c15">v2</font> &amp; <font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c2">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Maroon" title="c14">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp;  not <font color="Maroon" title="c15">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c14">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) or (  not <font color="Maroon" title="c14">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; <font color="Maroon" title="c15">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> &amp; (  not <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c15">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font></span>)</span> ) ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45_2_4_2"><span class="lab"><font color="Green" title="E47">A48</font></span></a>, <a class="txt" href="#E9:45_2_4_2"><span class="lab"><font color="Green" title="E48">A49</font></span></a>, <a class="txt" href="#E10:45_2_4_2"><span class="lab"><font color="Green" title="E49">A50</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E3:45"/><span class="lab"><font color="Green" title="E29">A51</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E28">A5</font></span></a>);<br/></span>
<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#V2" title="GLIB_000:attr.2">_finite</a>   <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a>  <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">H</font> being   <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</font>  ex <font color="Olive" title="b2">p</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="glib_000.html#V12" title="GLIB_000:attr.12">Graph-yielding</a>   <a href="glib_000.html#V14" title="GLIB_000:attr.14">_finite</a>   <a href="glpacy00.html#V2" title="GLPACY00:attr.2">Path-like</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Olive" title="b1">H</font> &amp; <font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">H</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; (  for <font color="Olive" title="b3">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b2">p</font>  st <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b4">v1</font>, <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">G</font> ex <font color="Olive" title="b6">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font>,<font color="Olive" title="b4">v1</font>,<font color="Olive" title="b6">e</font>,<font color="Olive" title="b5">v2</font> &amp; <font color="Olive" title="b6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Olive" title="b4">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> &amp;  not <font color="Olive" title="b5">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> &amp; (  not <font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b4">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> ) ) or (  not <font color="Olive" title="b4">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> &amp; <font color="Olive" title="b5">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> &amp; (  not <font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b5">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font></span>)</span> ) ) ) ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">H</font> be   <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="glib_000.html#V12" title="GLIB_000:attr.12">Graph-yielding</a>   <a href="glib_000.html#V14" title="GLIB_000:attr.14">_finite</a>   <a href="glpacy00.html#V2" title="GLPACY00:attr.2">Path-like</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">H</font> &amp; <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; (  for <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">p</font>  st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b3">v1</font>, <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">G</font> ex <font color="Olive" title="b5">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font>,<font color="Olive" title="b3">v1</font>,<font color="Olive" title="b5">e</font>,<font color="Olive" title="b4">v2</font> &amp; <font color="Olive" title="b5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Olive" title="b3">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> &amp;  not <font color="Olive" title="b4">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> &amp; (  not <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b3">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> ) ) or (  not <font color="Olive" title="b3">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> &amp; <font color="Olive" title="b4">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> &amp; (  not <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b4">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> ) ) ) ) ) )</span><br/>



<a NAME="E4:45"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T75" onmouseover="rs('glib_000/T75')" onmouseout="rh()">GLIB_000:75</a>, <a class="ref" href="nat_1.html#T21" onmouseover="rs('nat_1/T21')" onmouseout="rh()">NAT_1:21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:45"/>
 ex <font color="Olive" title="b1">p</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="glib_000.html#V12" title="GLIB_000:attr.12">Graph-yielding</a>   <a href="glib_000.html#V14" title="GLIB_000:attr.14">_finite</a>   <a href="glpacy00.html#V2" title="GLPACY00:attr.2">Path-like</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">H</font> &amp; <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; (  for <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">p</font>  st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b3">v1</font>, <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">G</font> ex <font color="Olive" title="b5">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> is    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font>,<font color="Olive" title="b3">v1</font>,<font color="Olive" title="b5">e</font>,<font color="Olive" title="b4">v2</font> &amp; <font color="Olive" title="b5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span></span>)</span> &amp; ( ( <font color="Olive" title="b3">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> &amp;  not <font color="Olive" title="b4">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> &amp; (  not <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b3">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> ) ) or (  not <font color="Olive" title="b3">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> &amp; <font color="Olive" title="b4">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> &amp; (  not <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b4">v2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> ) ) ) ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45"><span class="lab"><font color="Green" title="E29">A51</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
