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

<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="graph_1.html#NM3" title="GRAPH_1:NM.3">Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">sc</font> being  <a href="graph_2.html#V3" title="GRAPH_2:attr.3">simple</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c1">G</font> holds <br/> ( <font color="Olive" title="b1">sc</font> is   <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font> iff (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">sc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">sc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Olive" title="b1">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">sc</font> be   <a href="graph_2.html#V3" title="GRAPH_2:attr.3">simple</a>   <a href="graph_1.html#M2" title="GRAPH_1:mode.2">Chain</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">sc</font> is   <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font> iff (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">sc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">sc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 ) )</span><br/>



<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">sc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">sc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 ) implies <font color="Maroon" title="c2">sc</font> is   <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:77_1"/><span class="lab"><font color="Green" title="E57">A1</font></span>: 
<font color="Maroon" title="c2">sc</font> is   <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">sc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; not  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">sc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies  not <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 )</span><br/><span class="kw">assume </span><a NAME="E2:77_1"/><span class="lab"><font color="Green" title="E58">A2</font></span>: 
not  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">sc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">sc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies  not <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 )</span><br/><span class="kw">assume </span><a NAME="E3:77_1"/>
not  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">sc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span><br/><a NAME="E4:77_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">sc</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:77_1"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></span>;<br/><a NAME="E5:77_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A3</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 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="c2">sc</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">assume </span><a NAME="E6:77_1"/>
<font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">hence </span><a NAME="E7:77_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77_1"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="txt" href="#E5:77_1"><span class="lab"><font color="Green" title="E59">A3</font></span></a>, <a class="ref" href="graph_1.html#D14" title="GRAPH_1:def.14">GRAPH_1:def 14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:77"/><span class="lab"><font color="Green" title="E57">A4</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">sc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">sc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">sc</font> is   <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:77_2"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">sc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">sc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:77"><span class="lab"><font color="Green" title="E57">A4</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_2_1"><span class="kw">suppose </span></a><a NAME="E1:77_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">sc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">sc</font> is   <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font></span><br/><div class="add"><a NAME="E2:77_2_1"/><span class="kw">then </span>
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Olive" title="b2">m</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="c2">sc</font> holds <br/><font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font>
 ;<br/><span class="kw">hence </span><a NAME="E3:77_2_1"/>
<font color="Maroon" title="c2">sc</font> is   <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_1.html#D14" title="GRAPH_1:def.14">GRAPH_1:def 14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="77_2_2"><span class="kw">suppose </span></a><a NAME="E1:77_2_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">sc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">sc</font> is   <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font></span><br/><div class="add"><a NAME="E2:77_2_2"/><span class="kw">then </span>
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Olive" title="b2">m</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="c2">sc</font> holds <br/><font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:77_2_2"/>
<font color="Maroon" title="c2">sc</font> is   <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_1.html#D14" title="GRAPH_1:def.14">GRAPH_1:def 14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="77_2_3"><span class="kw">suppose </span></a><a NAME="E1:77_2_3"/><span class="lab"><font color="Green" title="E58">A5</font></span>: 
<font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">sc</font> is   <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_2_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font> &amp; <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">m</font> &amp; <font color="Maroon" title="c4">m</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="c2">sc</font> implies <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">m</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:77_2_3_1"/><span class="lab"><font color="Green" title="E59">A6</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">and </span><br/><a NAME="E2:77_2_3_1"/><span class="lab"><font color="Green" title="E60">A7</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">m</font>
 <span class="kw">and </span><br/><a NAME="E3:77_2_3_1"/><span class="lab"><font color="Green" title="E61">A8</font></span>: 
<font color="Maroon" title="c4">m</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="c2">sc</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">m</font></span><br/><span class="kw">thus </span><a NAME="E4:77_2_3_1"/>
<font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">m</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_2_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c5">vs</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E2:77_2_3_1_1"/><span class="lab"><font color="Green" title="E62">A9</font></span>: 
<font color="Maroon" title="c5">vs</font> <a href="graph_2.html#R1" title="GRAPH_2:pred.1">is_vertex_seq_of</a> <font color="Maroon" title="c2">sc</font>
 <span class="kw">and </span><br/><a NAME="E3:77_2_3_1_1"/><span class="lab"><font color="Green" title="E63">A10</font></span>: 
for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Olive" title="b2">m</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="c5">vs</font> &amp; <font color="Maroon" title="c5">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> holds <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b2">m</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="c5">vs</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_2.html#D10" target="_self" title="GRAPH_2:def.10">Def10</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">vm1</font> = <font color="Maroon" title="c5">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/>
<a NAME="E4:77_2_3_1_1"/><span class="lab"><font color="Green" title="E64">A11</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">vs</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="c2">sc</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:77_2_3_1_1"><span class="lab"><font color="Green" title="E62">A9</font></span></a>, <a class="ref" href="graph_2.html#D7" target="_self" title="GRAPH_2:def.7">Def7</a></span>;<br/>
<a NAME="E5:77_2_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A12</font></span>: 
<font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 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="c5">vs</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77_2_3_1"><span class="lab"><font color="Green" title="E61">A8</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E6:77_2_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A13</font></span>: 
<font color="Maroon" title="c5">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">vn1</font> = <font color="Maroon" title="c5">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">vn</font> = <font color="Maroon" title="c5">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">n</font>;<br/>
<a NAME="E7:77_2_3_1_1"/><span class="lab"><font color="Green" title="E67">A14</font></span>: 
<font color="Maroon" title="c3">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="c2">sc</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:77_2_3_1"><span class="lab"><font color="Green" title="E60">A7</font></span></a>, <a class="txt" href="#E3:77_2_3_1"><span class="lab"><font color="Green" title="E61">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E8:77_2_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A15</font></span>: 
<font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="graph_1.html#R2" title="GRAPH_1:pred.2">joins</a> <font color="Maroon" title="c5">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c5">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77_2_3_1"><span class="lab"><font color="Green" title="E59">A6</font></span></a>, <a class="txt" href="#E2:77_2_3_1_1"><span class="lab"><font color="Green" title="E62">A9</font></span></a>, <a class="ref" href="graph_2.html#D7" target="_self" title="GRAPH_2:def.7">Def7</a></span>;<br/>
<a NAME="E9:77_2_3_1_1"/>
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 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="c5">vs</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:77_2_3_1_1"><span class="lab"><font color="Green" title="E64">A11</font></span></a>, <a class="txt" href="#E7:77_2_3_1_1"><span class="lab"><font color="Green" title="E67">A14</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E10:77_2_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A16</font></span>: 
<font color="Maroon" title="c5">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E11:77_2_3_1_1"/>
<font color="Maroon" title="c3">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="c5">vs</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:77_2_3_1_1"><span class="lab"><font color="Green" title="E64">A11</font></span></a>, <a class="txt" href="#E7:77_2_3_1_1"><span class="lab"><font color="Green" title="E67">A14</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E12:77_2_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A17</font></span>: 
<font color="Maroon" title="c5">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77_2_3_1"><span class="lab"><font color="Green" title="E59">A6</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">vm</font> = <font color="Maroon" title="c5">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">m</font>;<br/>
<a NAME="E13:77_2_3_1_1"/><span class="lab"><font color="Green" title="E71">A18</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77_2_3_1"><span class="lab"><font color="Green" title="E59">A6</font></span></a>, <a class="txt" href="#E2:77_2_3_1"><span class="lab"><font color="Green" title="E60">A7</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E14:77_2_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A19</font></span>: 
<font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">m</font> <a href="graph_1.html#R2" title="GRAPH_1:pred.2">joins</a> <font color="Maroon" title="c5">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c5">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77_2_3_1"><span class="lab"><font color="Green" title="E61">A8</font></span></a>, <a class="txt" href="#E2:77_2_3_1_1"><span class="lab"><font color="Green" title="E62">A9</font></span></a>, <a class="ref" href="graph_2.html#D7" target="_self" title="GRAPH_2:def.7">Def7</a></span>;<br/>
<a NAME="E15:77_2_3_1_1"/><span class="lab"><font color="Green" title="E73">A20</font></span>: 
<font color="Maroon" title="c4">m</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="c5">vs</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77_2_3_1"><span class="lab"><font color="Green" title="E61">A8</font></span></a>, <a class="txt" href="#E4:77_2_3_1_1"><span class="lab"><font color="Green" title="E64">A11</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E16:77_2_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A21</font></span>: 
<font color="Maroon" title="c5">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">vs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:77_2_3_1_1"><span class="lab"><font color="Green" title="E71">A18</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<span class="kw">assume </span><a NAME="E17:77_2_3_1_1"/><span class="lab"><font color="Green" title="E75">A22</font></span>: 
not <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">sc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_2_3_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:77_2_3_1_1_1"/>
( ( <font color="Maroon" title="c5">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">m</font> &amp; <font color="Maroon" title="c5">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c5">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c5">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">m</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:77_2_3_1_1"><span class="lab"><font color="Green" title="E75">A22</font></span></a>, <a class="txt" href="#E8:77_2_3_1_1"><span class="lab"><font color="Green" title="E68">A15</font></span></a>, <a class="txt" href="#E14:77_2_3_1_1"><span class="lab"><font color="Green" title="E72">A19</font></span></a>, <a class="ref" href="graph_2.html#T32" target="_self" title="GRAPH_2:th.32">Th32</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_2_3_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:77_2_3_1_1_1_1"/><span class="lab"><font color="Green" title="E76">A23</font></span>: 
( <font color="Maroon" title="c5">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">m</font> &amp; <font color="Maroon" title="c5">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:77_2_3_1_1_1_1"/><span class="lab"><font color="Green" title="E77">A24</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:77_2_3_1"><span class="lab"><font color="Green" title="E60">A7</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E3:77_2_3_1_1_1_1"/>
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77_2_3_1"><span class="lab"><font color="Green" title="E59">A6</font></span></a>, <a class="txt" href="#E2:77_2_3_1"><span class="lab"><font color="Green" title="E60">A7</font></span></a>, <a class="txt" href="#E3:77_2_3_1_1"><span class="lab"><font color="Green" title="E63">A10</font></span></a>, <a class="txt" href="#E15:77_2_3_1_1"><span class="lab"><font color="Green" title="E73">A20</font></span></a>, <a class="txt" href="#E12:77_2_3_1_1"><span class="lab"><font color="Green" title="E70">A17</font></span></a>, <a class="txt" href="#E16:77_2_3_1_1"><span class="lab"><font color="Green" title="E74">A21</font></span></a>, <a class="txt" href="#E1:77_2_3_1_1_1_1"><span class="lab"><font color="Green" title="E76">A23</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:77_2_3_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77_2_3_1_1"><span class="lab"><font color="Green" title="E63">A10</font></span></a>, <a class="txt" href="#E5:77_2_3_1_1"><span class="lab"><font color="Green" title="E65">A12</font></span></a>, <a class="txt" href="#E10:77_2_3_1_1"><span class="lab"><font color="Green" title="E69">A16</font></span></a>, <a class="txt" href="#E6:77_2_3_1_1"><span class="lab"><font color="Green" title="E66">A13</font></span></a>, <a class="txt" href="#E1:77_2_3_1_1_1_1"><span class="lab"><font color="Green" title="E76">A23</font></span></a>, <a class="txt" href="#E2:77_2_3_1_1_1_1"><span class="lab"><font color="Green" title="E77">A24</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="77_2_3_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:77_2_3_1_1_1_2"/><span class="lab"><font color="Green" title="E76">A25</font></span>: 
( <font color="Maroon" title="c5">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c5">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">vs</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">m</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:77_2_3_1_1_1_2"/>
contradiction
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_2_3_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:77_2_3_1_1_1_2_1"/><span class="lab"><font color="Green" title="E77">A26</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:77_2_3_1"><span class="lab"><font color="Green" title="E60">A7</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E2:77_2_3_1_1_1_2_1"/><span class="lab"><font color="Green" title="E78">A27</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:77_2_3_1"><span class="lab"><font color="Green" title="E60">A7</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_2_3_1_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:77_2_3_1_1_1_2_1_1"/>
( <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> or <font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">m</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77_2_3_1_1_1_2_1"><span class="lab"><font color="Green" title="E77">A26</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_2_3_1_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:77_2_3_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E79">A28</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:77_2_3_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77_2_3_1"><span class="lab"><font color="Green" title="E59">A6</font></span></a>, <a class="txt" href="#E3:77_2_3_1_1"><span class="lab"><font color="Green" title="E63">A10</font></span></a>, <a class="txt" href="#E5:77_2_3_1_1"><span class="lab"><font color="Green" title="E65">A12</font></span></a>, <a class="txt" href="#E12:77_2_3_1_1"><span class="lab"><font color="Green" title="E70">A17</font></span></a>, <a class="txt" href="#E6:77_2_3_1_1"><span class="lab"><font color="Green" title="E66">A13</font></span></a>, <a class="txt" href="#E1:77_2_3_1_1_1_2"><span class="lab"><font color="Green" title="E76">A25</font></span></a>, <a class="txt" href="#E2:77_2_3_1_1_1_2_1"><span class="lab"><font color="Green" title="E78">A27</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:77_2_3_1_1_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77_2_3"><span class="lab"><font color="Green" title="E58">A5</font></span></a>, <a class="txt" href="#E17:77_2_3_1_1"><span class="lab"><font color="Green" title="E75">A22</font></span></a>, <a class="txt" href="#E1:77_2_3_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E79">A28</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="77_2_3_1_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:77_2_3_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E79">A29</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:77_2_3_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77_2_3_1"><span class="lab"><font color="Green" title="E59">A6</font></span></a>, <a class="txt" href="#E3:77_2_3_1_1"><span class="lab"><font color="Green" title="E63">A10</font></span></a>, <a class="txt" href="#E5:77_2_3_1_1"><span class="lab"><font color="Green" title="E65">A12</font></span></a>, <a class="txt" href="#E12:77_2_3_1_1"><span class="lab"><font color="Green" title="E70">A17</font></span></a>, <a class="txt" href="#E6:77_2_3_1_1"><span class="lab"><font color="Green" title="E66">A13</font></span></a>, <a class="txt" href="#E1:77_2_3_1_1_1_2"><span class="lab"><font color="Green" title="E76">A25</font></span></a>, <a class="txt" href="#E2:77_2_3_1_1_1_2_1"><span class="lab"><font color="Green" title="E78">A27</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:77_2_3_1_1_1_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77_2_3_1_1"><span class="lab"><font color="Green" title="E63">A10</font></span></a>, <a class="txt" href="#E15:77_2_3_1_1"><span class="lab"><font color="Green" title="E73">A20</font></span></a>, <a class="txt" href="#E10:77_2_3_1_1"><span class="lab"><font color="Green" title="E69">A16</font></span></a>, <a class="txt" href="#E16:77_2_3_1_1"><span class="lab"><font color="Green" title="E74">A21</font></span></a>, <a class="txt" href="#E1:77_2_3_1_1_1_2"><span class="lab"><font color="Green" title="E76">A25</font></span></a>, <a class="txt" href="#E1:77_2_3_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E79">A29</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:77_2_3"/>
<font color="Maroon" title="c2">sc</font> is   <a href="graph_1.html#NM6" title="GRAPH_1:NM.6">Path</a> of <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_1.html#D14" title="GRAPH_1:def.14">GRAPH_1:def 14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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>
