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<div class="add">

<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> ex <font color="Olive" title="b1">G</font> being   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a> ex <font color="Olive" title="b2">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b1">G</font> st <br/>( <font color="Olive" title="b1">G</font> is <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</a>  &amp; <font color="Olive" title="b1">G</font> is <font color="Maroon" title="c1">n</font> <a href="glib_013.html#V4" title="GLIB_013:attr.4">-edge</a>  &amp; <font color="Olive" title="b1">G</font> is  <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a>  &amp; (  not <font color="Olive" title="b1">G</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b2">v</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  ) );<br/>
<a NAME="E1:39_1_1"/><span class="lab"><font color="Green" title="E24">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="39_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span><font color="Maroon" title="c2">G</font> =  the   <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>   <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</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">  ex <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="c2">G</font> st <br/>( <font color="Maroon" title="c2">G</font> is <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</a>  &amp; <font color="Maroon" title="c2">G</font> is  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="glib_013.html#V4" title="GLIB_013:attr.4">-edge</a>  &amp; <font color="Maroon" title="c2">G</font> is  <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a>  &amp; (  not <font color="Maroon" title="c2">G</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b1">v</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  ) )</span><br/>

<span class="kw">take </span>
 the   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</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="c2">G</font> is <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</a>  &amp; <font color="Maroon" title="c2">G</font> is  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="glib_013.html#V4" title="GLIB_013:attr.4">-edge</a>  &amp; <font color="Maroon" title="c2">G</font> is  <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a>  &amp; (  not <font color="Maroon" title="c2">G</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies  the   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E1:39_1_1_1"/>
( <font color="Maroon" title="c2">G</font> is <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</a>  &amp; <font color="Maroon" title="c2">G</font> is  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="glib_013.html#V4" title="GLIB_013:attr.4">-edge</a>  &amp; <font color="Maroon" title="c2">G</font> is  <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a>  &amp; (  not <font color="Maroon" title="c2">G</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies  the   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:39_1_1"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
 for <font color="Olive" title="b1">m</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">m</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</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="39_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">m</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="c2">m</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:39_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">m</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="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">G2</font> being   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>, <font color="Maroon" title="c4">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G2</font><span class="kw"> such that </span><br/><a NAME="E3:39_1_1_2"/><span class="lab"><font color="Green" title="E26">A3</font></span>: 
( <font color="Maroon" title="c3">G2</font> is <font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</a>  &amp; <font color="Maroon" title="c3">G2</font> is <font color="Maroon" title="c2">m</font> <a href="glib_013.html#V4" title="GLIB_013:attr.4">-edge</a>  &amp; <font color="Maroon" title="c3">G2</font> is  <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a>  )
 <span class="kw">and </span><br/><a NAME="E4:39_1_1_2"/><span class="lab"><font color="Green" title="E27">A4</font></span>: 
(  not <font color="Maroon" title="c3">G2</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c4">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  )
 ;<br/>
<span class="kw">take </span><font color="Maroon" title="c5">G1</font> =  the    <a href="glib_006.html#M4" title="GLIB_006:mode.4">addAdjVertex</a> of <font color="Maroon" title="c3">G2</font>,<font color="Maroon" title="c4">v2</font>, <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c3">G2</font>, <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <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">G1</font> st <br/>( <font color="Maroon" title="c5">G1</font> is <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</a>  &amp; <font color="Maroon" title="c5">G1</font> is <font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V4" title="GLIB_013:attr.4">-edge</a>  &amp; <font color="Maroon" title="c5">G1</font> is  <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a>  &amp; (  not <font color="Maroon" title="c5">G1</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Olive" title="b1">v</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  ) )</span><br/>

<a NAME="E5:39_1_1_2"/><span class="lab"><font color="Green" title="E28">A5</font></span>: 
(  not  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c3">G2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c3">G2</font> &amp;  not  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font> )
 
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">v1</font> =  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c5">G1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39_1_1_2"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="ref" href="glib_006.html#T129" onmouseover="rs('glib_006/T129')" onmouseout="rh()">GLIB_006:129</a></span>;<br/>
<span class="kw">take </span>
<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="c5">G1</font> is <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</a>  &amp; <font color="Maroon" title="c5">G1</font> is <font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V4" title="GLIB_013:attr.4">-edge</a>  &amp; <font color="Maroon" title="c5">G1</font> is  <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a>  &amp; (  not <font color="Maroon" title="c5">G1</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c6">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  ) )</span><br/>

<font color="Maroon" title="c5">G1</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.order()</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">G2</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="#E5:39_1_1_2"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="ref" href="glib_006.html#T150" onmouseover="rs('glib_006/T150')" onmouseout="rh()">GLIB_006:150</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</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="#E3:39_1_1_2"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="glib_013.html#D3" onmouseover="rs('glib_013/D3')" onmouseout="rh()">GLIB_013:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1

;<br/>
<span class="kw">hence </span><a NAME="E8:39_1_1_2"/>
<font color="Maroon" title="c5">G1</font> is <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_013.html#D3" onmouseover="rs('glib_013/D3')" onmouseout="rh()">GLIB_013:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">G1</font> is <font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V4" title="GLIB_013:attr.4">-edge</a>  &amp; <font color="Maroon" title="c5">G1</font> is  <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a>  &amp; (  not <font color="Maroon" title="c5">G1</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c6">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  ) )</span><br/>

<font color="Maroon" title="c5">G1</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.size()</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">G2</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.size()</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="#E5:39_1_1_2"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="ref" href="glib_006.html#T150" onmouseover="rs('glib_006/T150')" onmouseout="rh()">GLIB_006:150</a></span>
<br/>.= 
<font color="Maroon" title="c2">m</font> <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="#E3:39_1_1_2"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="glib_013.html#D4" onmouseover="rs('glib_013/D4')" onmouseout="rh()">GLIB_013:def 4</a></span>
<br/>.= 
<font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1

;<br/>
<span class="kw">hence </span><a NAME="E10:39_1_1_2"/>
<font color="Maroon" title="c5">G1</font> is <font color="Maroon" title="c2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V4" title="GLIB_013:attr.4">-edge</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_013.html#D4" onmouseover="rs('glib_013/D4')" onmouseout="rh()">GLIB_013:def 4</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">G1</font> is  <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a>  &amp; (  not <font color="Maroon" title="c5">G1</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c6">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E11:39_1_1_2"/>
<font color="Maroon" title="c5">G1</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="#E3:39_1_1_2"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E4:39_1_1_2"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="ref" href="glpacy00.html#T10" target="_self" onmouseover="rs('glpacy00/T10')" onmouseout="rh()">Th18</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"> (  not <font color="Maroon" title="c5">G1</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c6">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E12:39_1_1_2"/>
 not <font color="Maroon" title="c5">G1</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> </span><br/>

<span class="kw">thus </span><a NAME="E13:39_1_1_2"/>
<font color="Maroon" title="c6">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39_1_1_2"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="ref" href="glib_006.html#T141" onmouseover="rs('glib_006/T141')" onmouseout="rh()">GLIB_006:141</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="E3:39_1_1"/>
 for <font color="Olive" title="b1">m</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">m</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="#E1:39_1_1"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E2:39_1_1"><span class="lab"><font color="Green" title="E25">A2</font></span></a>);<br/></span>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">G</font> being   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>, <font color="Maroon" title="c3">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="E5:39_1_1"/><span class="lab"><font color="Green" title="E26">A8</font></span>: 
( <font color="Maroon" title="c2">G</font> is <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</a>  &amp; <font color="Maroon" title="c2">G</font> is <font color="Maroon" title="c1">n</font> <a href="glib_013.html#V4" title="GLIB_013:attr.4">-edge</a>  &amp; <font color="Maroon" title="c2">G</font> is  <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a>  )
 <span class="kw">and </span><br/><a NAME="E6:39_1_1"/>
(  not <font color="Maroon" title="c2">G</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  implies <font color="Maroon" title="c3">v</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  )
 ;<br/>
<span class="kw">take </span>
<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="c2">G</font> is <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</a>  &amp; <font color="Maroon" title="c2">G</font> is <font color="Maroon" title="c1">n</font> <a href="glib_013.html#V4" title="GLIB_013:attr.4">-edge</a>  &amp; <font color="Maroon" title="c2">G</font> is  <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E7:39_1_1"/>
( <font color="Maroon" title="c2">G</font> is <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</a>  &amp; <font color="Maroon" title="c2">G</font> is <font color="Maroon" title="c1">n</font> <a href="glib_013.html#V4" title="GLIB_013:attr.4">-edge</a>  &amp; <font color="Maroon" title="c2">G</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="#E5:39_1_1"><span class="lab"><font color="Green" title="E26">A8</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>
