<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">P2</font> be   <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">v2</font>, <font color="Olive" title="b2">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <br/>  for <font color="Olive" title="b3">w2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">P2</font><br/>  for <font color="Olive" title="b4">P1</font> being    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c1">P2</font>,<font color="Olive" title="b1">v2</font>,<font color="Olive" title="b2">e</font>,<font color="Olive" title="b3">w2</font>  st ( <font color="Olive" title="b3">w2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  or <font color="Maroon" title="c1">P2</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) holds <br/><font color="Olive" title="b4">P1</font> is  <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">v2</font>, <font color="Maroon" title="c3">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">w2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">P2</font><br/>  for <font color="Olive" title="b2">P1</font> being    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c1">P2</font>,<font color="Maroon" title="c2">v2</font>,<font color="Maroon" title="c3">e</font>,<font color="Olive" title="b1">w2</font>  st ( <font color="Olive" title="b1">w2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  or <font color="Maroon" title="c1">P2</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) holds <br/><font color="Olive" title="b2">P1</font> is  <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">w2</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">P2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P1</font> being    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c1">P2</font>,<font color="Maroon" title="c2">v2</font>,<font color="Maroon" title="c3">e</font>,<font color="Maroon" title="c4">w2</font>  st ( <font color="Maroon" title="c4">w2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  or <font color="Maroon" title="c1">P2</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) holds <br/><font color="Olive" title="b1">P1</font> is  <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c5">P1</font> be    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c1">P2</font>,<font color="Maroon" title="c2">v2</font>,<font color="Maroon" title="c3">e</font>,<font color="Maroon" title="c4">w2</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">w2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  or <font color="Maroon" title="c1">P2</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) implies <font color="Maroon" title="c5">P1</font> is  <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a>  )</span><br/>







<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_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="E1:38_1"/>
( (  not <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">P2</font> &amp;  not <font color="Maroon" title="c2">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">P2</font> ) or <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">P2</font> or <font color="Maroon" title="c2">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">P2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1"><span class="kw">suppose </span></a><a NAME="E1:38_1_1"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
(  not <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">P2</font> &amp;  not <font color="Maroon" title="c2">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">P2</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">w2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  or <font color="Maroon" title="c1">P2</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) implies <font color="Maroon" title="c5">P1</font> is  <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a>  )</span><br/><div class="add"><span class="kw">assume </span><a NAME="E2:38_1_1"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
( <font color="Maroon" title="c4">w2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  or <font color="Maroon" title="c1">P2</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="c5">P1</font> is  <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a> </span><br/><span class="kw">thus </span><a NAME="E3:38_1_1"/>
<font color="Maroon" title="c5">P1</font> is  <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="glpacy00.html#D1" target="_self" onmouseover="rs('glpacy00/D1')" onmouseout="rh()">GLPACY00:def 1</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c5">P1</font> holds  <font color="Olive" title="b1">v</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> 2</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:38_1_1_1"/>
( <font color="Maroon" title="c4">w2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  or <font color="Maroon" title="c1">P2</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38_1_1"><span class="lab"><font color="Green" title="E25">A2</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:38_1_1_1_1"/><span class="lab"><font color="Green" title="E26">A3</font></span>: 
<font color="Maroon" title="c4">w2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c5">P1</font> holds  <font color="Olive" title="b1">v</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> 2</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c6">v1</font> = <font color="Maroon" title="c2">v2</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c5">P1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1_1"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="glib_006.html#T130" onmouseover="rs('glib_006/T130')" onmouseout="rh()">GLIB_006:130</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">w1</font> = <font color="Maroon" title="c4">w2</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c5">P1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_006.html#T68" onmouseover="rs('glib_006/T68')" onmouseout="rh()">GLIB_006:68</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c8">v</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c5">P1</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">v</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> 2</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:38_1_1_1_1_1"/>
( <font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">w1</font> or <font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</font> or ( <font color="Maroon" title="c8">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">v1</font> &amp; <font color="Maroon" title="c8">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">w1</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:38_1_1_1_1_1_1"/>
<font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">w1</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">v</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> 2</span><br/><div class="add"><span class="kw">then </span><font color="Maroon" title="c8">v</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">w2</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="#E1:38_1_1"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="glpacy00.html#T7" target="_self" onmouseover="rs('glpacy00/T7')" onmouseout="rh()">Th11</a></span>
<br/>.= 
1 <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1_1_1_1"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="glib_000.html#T174" onmouseover="rs('glib_000/T174')" onmouseout="rh()">GLIB_000:174</a></span>
<br/>.= 
2

;<br/><span class="kw">hence </span><a NAME="E3:38_1_1_1_1_1_1"/>
<font color="Maroon" title="c8">v</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:38_1_1_1_1_1_2"/>
<font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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"> <font color="Maroon" title="c8">v</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> 2</span><br/><div class="add"><a NAME="E2:38_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">v</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="#E1:38_1_1"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="glib_006.html#T142" onmouseover="rs('glib_006/T142')" onmouseout="rh()">GLIB_006:142</a>, <a class="ref" href="glib_000.html#T174" onmouseover="rs('glib_000/T174')" onmouseout="rh()">GLIB_000:174</a></span>;<br/><a NAME="E3:38_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">v</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> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T50" onmouseover="rs('card_1/T50')" onmouseout="rh()">CARD_1:50</a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E4:38_1_1_1_1_1_2"/>
<font color="Maroon" title="c8">v</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D2" onmouseover="rs('ordinal1/D2')" onmouseout="rh()">ORDINAL1:def 2</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="38_1_1_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:38_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E27">A4</font></span>: 
( <font color="Maroon" title="c8">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">v1</font> &amp; <font color="Maroon" title="c8">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">w1</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">v</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> 2</span><br/><div class="add"><a NAME="E2:38_1_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A5</font></span>: 
 not <font color="Maroon" title="c8">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v2</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="E3:38_1_1_1_1_1_3"/>
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">P2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v2</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="#E1:38_1_1"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="glib_006.html#D12" onmouseover="rs('glib_006/D12')" onmouseout="rh()">GLIB_006:def 12</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">v9</font> = <font color="Maroon" title="c8">v</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">P2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E5:38_1_1_1_1_1_3"/>
<font color="Maroon" title="c8">v</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v9</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="ref" href="glpacy00.html#T5" target="_self" onmouseover="rs('glpacy00/T5')" onmouseout="rh()">Th9</a></span>;<br/><span class="kw">hence </span><a NAME="E6:38_1_1_1_1_1_3"/>
<font color="Maroon" title="c8">v</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <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>; <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="38_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:38_1_1_1_2"/><span class="lab"><font color="Green" title="E26">A6</font></span>: 
<font color="Maroon" title="c1">P2</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">  for <font color="Olive" title="b1">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c5">P1</font> holds  <font color="Olive" title="b1">v</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> 2</span><br/><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">v</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c5">P1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">v</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> 2</span><br/><span class="lab"><font color="Green" title="E27">A7</font></span>: <font color="Maroon" title="c5">P1</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.order()</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">P2</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.order()</a></span>)</span> <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1_1"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="glib_006.html#T151" onmouseover="rs('glib_006/T151')" onmouseout="rh()">GLIB_006:151</a></span>
<br/>.= 
1 <a href="card_2.html#K1" title="CARD_2:func.1">+`</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1_1_1_2"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="ref" href="glib_000.html#T26" onmouseover="rs('glib_000/T26')" onmouseout="rh()">GLIB_000:26</a></span>
<br/>.= 
2

;<br/><a NAME="E3:38_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A8</font></span>: 
<font color="Maroon" title="c5">P1</font> is  <a href="glib_013.html#V1" title="GLIB_013:attr.1">vertex-finite</a> 
 ;<br/><a NAME="E4:38_1_1_1_2"/>
 not <font color="Maroon" title="c5">P1</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="#E2:38_1_1_1_2"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="ref" href="glib_000.html#T26" onmouseover="rs('glib_000/T26')" onmouseout="rh()">GLIB_000:26</a></span>;<br/><a NAME="E5:38_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A9</font></span>: 
<font color="Maroon" title="c6">v</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="#E2:38_1_1_1_2"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="txt" href="#E3:38_1_1_1_2"><span class="lab"><font color="Green" title="E28">A8</font></span></a>, <a class="ref" href="glib_008.html#T27" onmouseover="rs('glib_008/T27')" onmouseout="rh()">GLIB_008:27</a>, <a class="ref" href="glib_000.html#T174" onmouseover="rs('glib_000/T174')" onmouseout="rh()">GLIB_000:174</a></span>;<br/><a NAME="E6:38_1_1_1_2"/>
1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E7:38_1_1_1_2"/>
<font color="Maroon" title="c6">v</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:38_1_1_1_2"><span class="lab"><font color="Green" title="E29">A9</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2"><span class="kw">suppose </span></a><a NAME="E1:38_1_2"/>
( <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">P2</font> or <font color="Maroon" title="c2">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">P2</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">w2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  or <font color="Maroon" title="c1">P2</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) implies <font color="Maroon" title="c5">P1</font> is  <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a>  )</span><br/><div class="add"><a NAME="E2:38_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">P1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c1">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_006.html#D12" onmouseover="rs('glib_006/D12')" onmouseout="rh()">GLIB_006:def 12</a></span>;<br/><span class="kw">hence </span><a NAME="E3:38_1_2"/>
( ( <font color="Maroon" title="c4">w2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  or <font color="Maroon" title="c1">P2</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  ) implies <font color="Maroon" title="c5">P1</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="glpacy00.html#E21"><span class="lab"><font color="Green" title="E21">Lm3</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>
