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

<b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:21"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">f</font> is <a href="jgraph_1.html#V1" target="_self">nodic</a> &amp;  <a href="jgraph_1.html#K4" target="_self">PairF</a> <font color="Maroon">f</font> is <a href="graph_4.html#V1">Simple</a> )
 ;<br/>

<b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> as    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <span class="p1">(<span class="default"><a href="jgraph_1.html#K1" target="_self">PGraph</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1"><b>suppose </b></a><a NAME="E1:21_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/><div class="add"><a NAME="E2:21_1_1"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">f1</font> <a href="graph_4.html#R3">is_oriented_vertex_seq_of</a>  <a href="jgraph_1.html#K4" target="_self">PairF</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/><a NAME="E3:21_1_1"/>
for <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">j</font> &amp; ( ( <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> ) or <font color="Olive">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> ) holds <br/> <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">i</font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:21_1_1_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> &amp; ( ( <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> ) or <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> ) )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1_1_1_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 or <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:21_1_1_1_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/><div class="add"><a NAME="E2:21_1_1_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:21_1_1_1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:21_1_1_1_1_1"/><i><font color="Green">E40</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:21_1_1_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:21_1_1_1_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:21_1_1_1_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:21_1_1_1_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E9:21_1_1_1_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E10:21_1_1_1_1_1"/><i><font color="Green">E63</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E11:21_1_1_1_1_1"/><i><font color="Green">E64</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1_1_1_1_1_1"/><i><font color="Green">E65</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1_1_1_1_1_1_1_1"/>
( ( <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; ( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> or <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font> ) or ( <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; ( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> or <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font> ) or  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font> )
 </span><b>by </b><i>, , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:21_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E66</font></i>: 
( <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; ( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> or <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1_1_1_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> or <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 </span><b>by </b><i><a class="ref" href="jgraph_1.html#T2" target="_self">Th2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1_1_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:21_1_1_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_1_1_1_1_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, , , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1_1_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:21_1_1_1_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_1_1_1_1_1_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, , , , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:21_1_1_1_1_1_1_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:21_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E67</font></i>: 
( <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; ( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> or <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) &amp;  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1_1_1_1_1_1_1_1_2_1_1"/>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> or <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 </span><b>by </b><i><a class="ref" href="jgraph_1.html#T2" target="_self">Th2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1_1_1_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:21_1_1_1_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_1_1_1_1_1_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i>, , , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1_1_1_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:21_1_1_1_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_1_1_1_1_1_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i>, , , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:21_1_1_1_1_1_1_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1_1_1_1_1_3"><b>case </b></a><a NAME="E1:21_1_1_1_1_1_1_1_1_3"/>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:21_1_1_1_1_1_1_1_1_3"/><b>then </b>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font>
 <b>by </b><i>, , <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E3:21_1_1_1_1_1_1_1_1_3"/><b>then </b><i><font color="Green">E68</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E4:21_1_1_1_1_1_1_1_1_3"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E5:21_1_1_1_1_1_1_1_1_3"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E6:21_1_1_1_1_1_1_1_1_3"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><a NAME="E7:21_1_1_1_1_1_1_1_1_3"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_1_1_1_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1_1_1_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1_1_1_1_1_1_1_1_3_1_1"/>
( ( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> ) )
 </span><b>by </b><i><a class="ref" href="jgraph_1.html#T2" target="_self">Th2</a>, <a class="txt" href="jgraph_1.html#E1:21"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jgraph_1.html#E2:21_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jgraph_1.html#E1:21_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jgraph_1.html#E1:21_1_1_1_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="sppol_1.html#T25">SPPOL_1:25</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1_1_1_1_1_3_1_1_1"><b>case </b></a><a NAME="E1:21_1_1_1_1_1_1_1_1_3_1_1_1"/>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_1_1_1_1_1_1_1_3_1_1_1"/>
contradiction
 <b>by </b><i>, , , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1_1_1_1_1_3_1_1_2"><b>case </b></a><a NAME="E1:21_1_1_1_1_1_1_1_1_3_1_1_2"/>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_1_1_1_1_1_1_1_3_1_1_2"/>
contradiction
 <b>by </b><i>, , , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:21_1_1_1_1_1_1_1_1_3"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:21_1_1_1_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:21_1_1_1_1_1"/>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:21_1_1_1_1_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><div class="add"><a NAME="E2:21_1_1_1_1_2"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E3:21_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>hence </b><a NAME="E4:21_1_1_1_1_2"/>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E4:21_1_1"/>
<font color="Maroon">f</font> is <a href="goboard5.html#V1">s.c.c.</a>
 <b>by </b><i><a class="ref" href="goboard5.html#D4">GOBOARD5:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2"><b>suppose </b></a><a NAME="E1:21_1_2"/><i><font color="Green">E73</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><div class="add"><a NAME="E2:21_1_2"/>
for <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">j</font> &amp; ( ( <font color="Olive">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> ) or <font color="Olive">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> ) holds <br/> <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">i</font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Olive">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:21_1_2_1"/>
( <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> &amp; ( ( <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> ) or <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> ) )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1_2_1_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 or <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:21_1_2_1_1_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 ;<br/><div class="add"><a NAME="E2:21_1_2_1_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:21_1_2_1_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:21_1_2_1_1_1"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E5:21_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>hence </b><a NAME="E6:21_1_2_1_1_1"/>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:21_1_2_1_1_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><div class="add"><a NAME="E2:21_1_2_1_1_2"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E3:21_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>hence </b><a NAME="E4:21_1_2_1_1_2"/>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">i</font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E3:21_1_2"/>
<font color="Maroon">f</font> is <a href="goboard5.html#V1">s.c.c.</a>
 <b>by </b><i><a class="ref" href="goboard5.html#D4">GOBOARD5:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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