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

<b>let </b><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c3">G</font> be   <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a>;<br/><b>let </b><font color="Maroon" title="c4">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:23"/><i><font color="Green" title="E23">A1</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i</font>
 <b>and </b><br/><a NAME="E2:23"/><i><font color="Green" title="E24">A2</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">G</font>
 <b>and </b><br/><a NAME="E3:23"/><i><font color="Green" title="E25">A3</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">j</font>
 <b>and </b><br/><a NAME="E4:23"/><i><font color="Green" title="E26">A4</font></i>: 
<font color="Maroon" title="c2">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">G</font>
 <b>and </b><br/><a NAME="E5:23"/><i><font color="Green" title="E27">A5</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K2" title="GOBRD13:func.2">Values</a> <font color="Maroon" title="c3">G</font>
 <b>and </b><br/><a NAME="E6:23"/><i><font color="Green" title="E28">A6</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>
 ;<br/>

<a NAME="E7:23"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font> &amp; not <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> holds <br/><font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:23_1"/><i><font color="Green" title="E29">A7</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>
 <b>and </b><br/><a NAME="E2:23_1"/><i><font color="Green" title="E30">A8</font></i>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>
 ;<br/>

<b>assume </b><a NAME="E3:23_1"/><i><font color="Green" title="E31">A9</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font> )
 ;<br/>

<a NAME="E4:23_1"/><i><font color="Green" title="E32">A10</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> )
 
<b>by </b><i><a class="ref" href="topreal1.html#T6" title="TOPREAL1:th.6">TOPREAL1:6</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1"/>
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font> or <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> or <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> or <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font> )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E1:23"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="jordan9.html#E2:23"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="jordan9.html#E3:23"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="jordan9.html#E4:23"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="jordan9.html#E5:23"><i><font color="Green" title="E27">A5</font></i></a>, <a class="txt" href="jordan9.html#E6:23"><i><font color="Green" title="E28">A6</font></i></a>, <a class="ref" href="jordan9.html#T21" target="_self" title="JORDAN9:th.21">Th21</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1"><b>suppose </b></a><a NAME="E1:23_1_1_1"/><i><font color="Green" title="E33">A11</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>
 ;<br/><div class="add"><a NAME="E2:23_1_1_1"/><b>then </b><i><font color="Green" title="E34">A12</font></i>: 
( <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:23"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="jordan9.html#E2:23"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="jordan9.html#E3:23"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="jordan9.html#E4:23"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="jordan9.html#E2:23_1"><i><font color="Green" title="E30">A8</font></i></a>, <a class="txt" href="jordan9.html#E4:23_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="ref" href="jordan9.html#T19" target="_self" title="JORDAN9:th.19">Th19</a></i>;<br/><a NAME="E3:23_1_1_1"/><i><font color="Green" title="E35">A13</font></i>: 
( <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:23"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="jordan9.html#E2:23"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="jordan9.html#E3:23"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="jordan9.html#E4:23"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="jordan9.html#E2:23_1"><i><font color="Green" title="E30">A8</font></i></a>, <a class="txt" href="jordan9.html#E4:23_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="txt" href="jordan9.html#E1:23_1_1_1"><i><font color="Green" title="E33">A11</font></i></a>, <a class="ref" href="jordan9.html#T19" target="_self" title="JORDAN9:th.19">Th19</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_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="E1:23_1_1_1_1_1"/>
( ( <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) or ( <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) or ( <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) or ( <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:23_1_1_1_1_1_1"/>
( <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:23_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:23_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a>, <a class="ref" href="topreal1.html#T10" title="TOPREAL1:th.10">TOPREAL1:10</a></i>;<br/><a NAME="E3:23_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:23_1_1_1"><i><font color="Green" title="E34">A12</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:23_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E3:23_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:23_1_1_1_1_1_2"/>
( <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:23_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:23_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a>, <a class="ref" href="topreal1.html#T10" title="TOPREAL1:th.10">TOPREAL1:10</a></i>;<br/><a NAME="E3:23_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E36">A14</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:23_1_1_1"><i><font color="Green" title="E34">A12</font></i></a>, <a class="txt" href="jordan9.html#E3:23_1_1_1"><i><font color="Green" title="E35">A13</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E4:23_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan9.html#E3:23_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></i>;<br/><a NAME="E5:23_1_1_1_1_1_2"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> is <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:23_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="jordan9.html#E4:23_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="txt" href="jordan9.html#E3:23_1_1_1_1_1_2"><i><font color="Green" title="E36">A14</font></i></a>, <a class="ref" href="sppol_1.html#T39" title="SPPOL_1:th.39">SPPOL_1:39</a></i>;<br/><a NAME="E6:23_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="ref" href="sppol_1.html#T37" title="SPPOL_1:th.37">SPPOL_1:37</a></i>;<br/><b>hence </b><a NAME="E7:23_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E3:23_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="txt" href="jordan9.html#E3:23_1_1_1_1_1_2"><i><font color="Green" title="E36">A14</font></i></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:23_1_1_1_1_1_3"/>
( <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:23_1_1_1_1_1_3"/><b>then </b>
( <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:23_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a>, <a class="ref" href="topreal1.html#T10" title="TOPREAL1:th.10">TOPREAL1:10</a></i>;<br/><a NAME="E3:23_1_1_1_1_1_3"/><b>then </b><i><font color="Green" title="E36">A15</font></i>: 
( <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:23_1_1_1"><i><font color="Green" title="E34">A12</font></i></a>, <a class="txt" href="jordan9.html#E3:23_1_1_1"><i><font color="Green" title="E35">A13</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E4:23_1_1_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan9.html#E3:23_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></i>;<br/><a NAME="E5:23_1_1_1_1_1_3"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> is <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:23_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="jordan9.html#E4:23_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="txt" href="jordan9.html#E3:23_1_1_1_1_1_3"><i><font color="Green" title="E36">A15</font></i></a>, <a class="ref" href="sppol_1.html#T39" title="SPPOL_1:th.39">SPPOL_1:39</a></i>;<br/><a NAME="E6:23_1_1_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="ref" href="sppol_1.html#T37" title="SPPOL_1:th.37">SPPOL_1:37</a></i>;<br/><b>hence </b><a NAME="E7:23_1_1_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E3:23_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="txt" href="jordan9.html#E3:23_1_1_1_1_1_3"><i><font color="Green" title="E36">A15</font></i></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:23_1_1_1_1_1_4"/>
( <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:23_1_1_1_1_1_4"/><b>then </b>
( <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:23_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a>, <a class="ref" href="topreal1.html#T10" title="TOPREAL1:th.10">TOPREAL1:10</a></i>;<br/><a NAME="E3:23_1_1_1_1_1_4"/><b>then </b>
( <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:23_1_1_1"><i><font color="Green" title="E35">A13</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:23_1_1_1_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E3:23_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:23_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_2"><b>suppose </b></a><a NAME="E1:23_1_1_2"/><i><font color="Green" title="E33">A16</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:23_1_1_2"/><b>then </b><i><font color="Green" title="E34">A17</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan9.html#E1:23"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="jordan9.html#E2:23"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="jordan9.html#E3:23"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="jordan9.html#E4:23"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="jordan9.html#T18" target="_self" title="JORDAN9:th.18">Th18</a></i>;<br/><a NAME="E3:23_1_1_2"/><b>then </b><i><font color="Green" title="E35">A18</font></i>: 
( <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:23"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="jordan9.html#E2:23"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="jordan9.html#E3:23"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="jordan9.html#E4:23"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="jordan9.html#E2:23_1"><i><font color="Green" title="E30">A8</font></i></a>, <a class="txt" href="jordan9.html#E4:23_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="txt" href="jordan9.html#E1:23_1_1_2"><i><font color="Green" title="E33">A16</font></i></a>, <a class="ref" href="jordan9.html#T19" target="_self" title="JORDAN9:th.19">Th19</a></i>;<br/><a NAME="E4:23_1_1_2"/><i><font color="Green" title="E36">A19</font></i>: 
( <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:23"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="jordan9.html#E2:23"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="jordan9.html#E3:23"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="jordan9.html#E4:23"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="jordan9.html#E2:23_1"><i><font color="Green" title="E30">A8</font></i></a>, <a class="txt" href="jordan9.html#E4:23_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="txt" href="jordan9.html#E1:23_1_1_2"><i><font color="Green" title="E33">A16</font></i></a>, <a class="txt" href="jordan9.html#E2:23_1_1_2"><i><font color="Green" title="E34">A17</font></i></a>, <a class="ref" href="jordan9.html#T19" target="_self" title="JORDAN9:th.19">Th19</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_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="E1:23_1_1_2_1_1"/>
( ( <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) or ( <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) or ( <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) or ( <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:23_1_1_2_1_1_1"/>
( <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:23_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:23_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a>, <a class="ref" href="topreal1.html#T10" title="TOPREAL1:th.10">TOPREAL1:10</a></i>;<br/><a NAME="E3:23_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E37">A20</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:23_1_1_2"><i><font color="Green" title="E35">A18</font></i></a>, <a class="txt" href="jordan9.html#E4:23_1_1_2"><i><font color="Green" title="E36">A19</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E4:23_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan9.html#E3:23_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></i>;<br/><a NAME="E5:23_1_1_2_1_1_1"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> is <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:23_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="jordan9.html#E4:23_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="txt" href="jordan9.html#E3:23_1_1_2_1_1_1"><i><font color="Green" title="E37">A20</font></i></a>, <a class="ref" href="sppol_1.html#T39" title="SPPOL_1:th.39">SPPOL_1:39</a></i>;<br/><a NAME="E6:23_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="ref" href="sppol_1.html#T37" title="SPPOL_1:th.37">SPPOL_1:37</a></i>;<br/><b>hence </b><a NAME="E7:23_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E3:23_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="txt" href="jordan9.html#E3:23_1_1_2_1_1_1"><i><font color="Green" title="E37">A20</font></i></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:23_1_1_2_1_1_2"/>
( <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:23_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:23_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a>, <a class="ref" href="topreal1.html#T10" title="TOPREAL1:th.10">TOPREAL1:10</a></i>;<br/><a NAME="E3:23_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:23_1_1_2"><i><font color="Green" title="E35">A18</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:23_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E3:23_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:23_1_1_2_1_1_3"/>
( <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:23_1_1_2_1_1_3"/><b>then </b>
( <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:23_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a>, <a class="ref" href="topreal1.html#T10" title="TOPREAL1:th.10">TOPREAL1:10</a></i>;<br/><a NAME="E3:23_1_1_2_1_1_3"/><b>then </b>
( <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E4:23_1_1_2"><i><font color="Green" title="E36">A19</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:23_1_1_2_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E3:23_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_2_1_1_4"><b>suppose </b></a><a NAME="E1:23_1_1_2_1_1_4"/>
( <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:23_1_1_2_1_1_4"/><b>then </b>
( <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:23_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a>, <a class="ref" href="topreal1.html#T10" title="TOPREAL1:th.10">TOPREAL1:10</a></i>;<br/><a NAME="E3:23_1_1_2_1_1_4"/><b>then </b><i><font color="Green" title="E37">A21</font></i>: 
( <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:23_1_1_2"><i><font color="Green" title="E35">A18</font></i></a>, <a class="txt" href="jordan9.html#E4:23_1_1_2"><i><font color="Green" title="E36">A19</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E4:23_1_1_2_1_1_4"/><b>then </b>
<font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan9.html#E3:23_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></i>;<br/><a NAME="E5:23_1_1_2_1_1_4"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> is <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:23_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="jordan9.html#E4:23_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="txt" href="jordan9.html#E3:23_1_1_2_1_1_4"><i><font color="Green" title="E37">A21</font></i></a>, <a class="ref" href="sppol_1.html#T39" title="SPPOL_1:th.39">SPPOL_1:39</a></i>;<br/><a NAME="E6:23_1_1_2_1_1_4"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="ref" href="sppol_1.html#T37" title="SPPOL_1:th.37">SPPOL_1:37</a></i>;<br/><b>hence </b><a NAME="E7:23_1_1_2_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E3:23_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="txt" href="jordan9.html#E3:23_1_1_2_1_1_4"><i><font color="Green" title="E37">A21</font></i></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:23_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3"><b>suppose </b></a><a NAME="E1:23_1_1_3"/>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:23_1_1_3"/><b>then </b><i><font color="Green" title="E33">A22</font></i>: 
( <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:23"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="jordan9.html#E2:23"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="jordan9.html#E3:23"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="jordan9.html#E4:23"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="jordan9.html#T18" target="_self" title="JORDAN9:th.18">Th18</a></i>;<br/><a NAME="E3:23_1_1_3"/><b>then </b><i><font color="Green" title="E34">A23</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:23"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="jordan9.html#E2:23"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="jordan9.html#E3:23"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="jordan9.html#E4:23"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="jordan9.html#E2:23_1"><i><font color="Green" title="E30">A8</font></i></a>, <a class="txt" href="jordan9.html#E4:23_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="ref" href="jordan9.html#T19" target="_self" title="JORDAN9:th.19">Th19</a></i>;<br/><a NAME="E4:23_1_1_3"/><i><font color="Green" title="E35">A24</font></i>: 
( <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:23"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="jordan9.html#E2:23"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="jordan9.html#E3:23"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="jordan9.html#E4:23"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="jordan9.html#E2:23_1"><i><font color="Green" title="E30">A8</font></i></a>, <a class="txt" href="jordan9.html#E4:23_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="txt" href="jordan9.html#E2:23_1_1_3"><i><font color="Green" title="E33">A22</font></i></a>, <a class="ref" href="jordan9.html#T19" target="_self" title="JORDAN9:th.19">Th19</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_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="E1:23_1_1_3_1_1"/>
( ( <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) or ( <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) or ( <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) or ( <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:23_1_1_3_1_1_1"/>
( <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:23_1_1_3_1_1_1"/><b>then </b>
( <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:23_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a>, <a class="ref" href="topreal1.html#T10" title="TOPREAL1:th.10">TOPREAL1:10</a></i>;<br/><a NAME="E3:23_1_1_3_1_1_1"/><b>then </b>
( <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E4:23_1_1_3"><i><font color="Green" title="E35">A24</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:23_1_1_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E3:23_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:23_1_1_3_1_1_2"/>
( <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:23_1_1_3_1_1_2"/><b>then </b>
( <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:23_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a>, <a class="ref" href="topreal1.html#T10" title="TOPREAL1:th.10">TOPREAL1:10</a></i>;<br/><a NAME="E3:23_1_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E36">A25</font></i>: 
( <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:23_1_1_3"><i><font color="Green" title="E34">A23</font></i></a>, <a class="txt" href="jordan9.html#E4:23_1_1_3"><i><font color="Green" title="E35">A24</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E4:23_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan9.html#E3:23_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></i>;<br/><a NAME="E5:23_1_1_3_1_1_2"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> is <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:23_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="jordan9.html#E4:23_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="txt" href="jordan9.html#E3:23_1_1_3_1_1_2"><i><font color="Green" title="E36">A25</font></i></a>, <a class="ref" href="sppol_1.html#T39" title="SPPOL_1:th.39">SPPOL_1:39</a></i>;<br/><a NAME="E6:23_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="ref" href="sppol_1.html#T37" title="SPPOL_1:th.37">SPPOL_1:37</a></i>;<br/><b>hence </b><a NAME="E7:23_1_1_3_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E3:23_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="txt" href="jordan9.html#E3:23_1_1_3_1_1_2"><i><font color="Green" title="E36">A25</font></i></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:23_1_1_3_1_1_3"/>
( <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:23_1_1_3_1_1_3"/><b>then </b>
( <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:23_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a>, <a class="ref" href="topreal1.html#T10" title="TOPREAL1:th.10">TOPREAL1:10</a></i>;<br/><a NAME="E3:23_1_1_3_1_1_3"/><b>then </b><i><font color="Green" title="E36">A26</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:23_1_1_3"><i><font color="Green" title="E34">A23</font></i></a>, <a class="txt" href="jordan9.html#E4:23_1_1_3"><i><font color="Green" title="E35">A24</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E4:23_1_1_3_1_1_3"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan9.html#E3:23_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></i>;<br/><a NAME="E5:23_1_1_3_1_1_3"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> is <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:23_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="jordan9.html#E4:23_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="txt" href="jordan9.html#E3:23_1_1_3_1_1_3"><i><font color="Green" title="E36">A26</font></i></a>, <a class="ref" href="sppol_1.html#T39" title="SPPOL_1:th.39">SPPOL_1:39</a></i>;<br/><a NAME="E6:23_1_1_3_1_1_3"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="ref" href="sppol_1.html#T37" title="SPPOL_1:th.37">SPPOL_1:37</a></i>;<br/><b>hence </b><a NAME="E7:23_1_1_3_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E3:23_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="txt" href="jordan9.html#E3:23_1_1_3_1_1_3"><i><font color="Green" title="E36">A26</font></i></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3_1_1_4"><b>suppose </b></a><a NAME="E1:23_1_1_3_1_1_4"/>
( <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:23_1_1_3_1_1_4"/><b>then </b>
( <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:23_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a>, <a class="ref" href="topreal1.html#T10" title="TOPREAL1:th.10">TOPREAL1:10</a></i>;<br/><a NAME="E3:23_1_1_3_1_1_4"/><b>then </b>
( <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:23_1_1_3"><i><font color="Green" title="E34">A23</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:23_1_1_3_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E3:23_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:23_1_1_3"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_4"><b>suppose </b></a><a NAME="E1:23_1_1_4"/><i><font color="Green" title="E33">A27</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c2">j</font>
 ;<br/><div class="add"><a NAME="E2:23_1_1_4"/><b>then </b><i><font color="Green" title="E34">A28</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan9.html#E1:23"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="jordan9.html#E2:23"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="jordan9.html#E3:23"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="jordan9.html#E4:23"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="jordan9.html#T18" target="_self" title="JORDAN9:th.18">Th18</a></i>;<br/><a NAME="E3:23_1_1_4"/><b>then </b><i><font color="Green" title="E35">A29</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:23"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="jordan9.html#E2:23"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="jordan9.html#E3:23"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="jordan9.html#E4:23"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="jordan9.html#E2:23_1"><i><font color="Green" title="E30">A8</font></i></a>, <a class="txt" href="jordan9.html#E4:23_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="txt" href="jordan9.html#E1:23_1_1_4"><i><font color="Green" title="E33">A27</font></i></a>, <a class="ref" href="jordan9.html#T19" target="_self" title="JORDAN9:th.19">Th19</a></i>;<br/><a NAME="E4:23_1_1_4"/><i><font color="Green" title="E36">A30</font></i>: 
( <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:23"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="jordan9.html#E2:23"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="jordan9.html#E3:23"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="jordan9.html#E4:23"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="jordan9.html#E2:23_1"><i><font color="Green" title="E30">A8</font></i></a>, <a class="txt" href="jordan9.html#E4:23_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="txt" href="jordan9.html#E1:23_1_1_4"><i><font color="Green" title="E33">A27</font></i></a>, <a class="txt" href="jordan9.html#E2:23_1_1_4"><i><font color="Green" title="E34">A28</font></i></a>, <a class="ref" href="jordan9.html#T19" target="_self" title="JORDAN9:th.19">Th19</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1_4_1_1"/>
( ( <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) or ( <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) or ( <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) or ( <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:23_1_1_4_1_1_1"/>
( <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:23_1_1_4_1_1_1"/><b>then </b>
( <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:23_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a>, <a class="ref" href="topreal1.html#T10" title="TOPREAL1:th.10">TOPREAL1:10</a></i>;<br/><a NAME="E3:23_1_1_4_1_1_1"/><b>then </b><i><font color="Green" title="E37">A31</font></i>: 
( <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:23_1_1_4"><i><font color="Green" title="E35">A29</font></i></a>, <a class="txt" href="jordan9.html#E4:23_1_1_4"><i><font color="Green" title="E36">A30</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E4:23_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan9.html#E3:23_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></i>;<br/><a NAME="E5:23_1_1_4_1_1_1"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> is <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:23_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="jordan9.html#E4:23_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="txt" href="jordan9.html#E3:23_1_1_4_1_1_1"><i><font color="Green" title="E37">A31</font></i></a>, <a class="ref" href="sppol_1.html#T39" title="SPPOL_1:th.39">SPPOL_1:39</a></i>;<br/><a NAME="E6:23_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="ref" href="sppol_1.html#T37" title="SPPOL_1:th.37">SPPOL_1:37</a></i>;<br/><b>hence </b><a NAME="E7:23_1_1_4_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E3:23_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="txt" href="jordan9.html#E3:23_1_1_4_1_1_1"><i><font color="Green" title="E37">A31</font></i></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:23_1_1_4_1_1_2"/>
( <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:23_1_1_4_1_1_2"/><b>then </b>
( <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:23_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a>, <a class="ref" href="topreal1.html#T10" title="TOPREAL1:th.10">TOPREAL1:10</a></i>;<br/><a NAME="E3:23_1_1_4_1_1_2"/><b>then </b>
( <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E4:23_1_1_4"><i><font color="Green" title="E36">A30</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:23_1_1_4_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E3:23_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_4_1_1_3"><b>suppose </b></a><a NAME="E1:23_1_1_4_1_1_3"/>
( <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:23_1_1_4_1_1_3"/><b>then </b>
( <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:23_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a>, <a class="ref" href="topreal1.html#T10" title="TOPREAL1:th.10">TOPREAL1:10</a></i>;<br/><a NAME="E3:23_1_1_4_1_1_3"/><b>then </b>
( <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:23_1_1_4"><i><font color="Green" title="E35">A29</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:23_1_1_4_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E3:23_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_4_1_1_4"><b>suppose </b></a><a NAME="E1:23_1_1_4_1_1_4"/>
( <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 ;<br/><div class="add"><a NAME="E2:23_1_1_4_1_1_4"/><b>then </b>
( <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:23_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="topreal1.html#T9" title="TOPREAL1:th.9">TOPREAL1:9</a>, <a class="ref" href="topreal1.html#T10" title="TOPREAL1:th.10">TOPREAL1:10</a></i>;<br/><a NAME="E3:23_1_1_4_1_1_4"/><b>then </b><i><font color="Green" title="E37">A32</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c6">b</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:23_1_1_4"><i><font color="Green" title="E35">A29</font></i></a>, <a class="txt" href="jordan9.html#E4:23_1_1_4"><i><font color="Green" title="E36">A30</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E4:23_1_1_4_1_1_4"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan9.html#E3:23_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></i>;<br/><a NAME="E5:23_1_1_4_1_1_4"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> is <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:23_1"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="jordan9.html#E4:23_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="txt" href="jordan9.html#E3:23_1_1_4_1_1_4"><i><font color="Green" title="E37">A32</font></i></a>, <a class="ref" href="sppol_1.html#T39" title="SPPOL_1:th.39">SPPOL_1:39</a></i>;<br/><a NAME="E6:23_1_1_4_1_1_4"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="ref" href="sppol_1.html#T37" title="SPPOL_1:th.37">SPPOL_1:37</a></i>;<br/><b>hence </b><a NAME="E7:23_1_1_4_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E3:23_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="txt" href="jordan9.html#E3:23_1_1_4_1_1_4"><i><font color="Green" title="E37">A32</font></i></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:23_1_1_4"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E8:23"/>
<font color="Maroon" title="c4">p</font> <a href="sppol_1.html#R1" title="SPPOL_1:pred.1">is_extremal_in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E6:23"><i><font color="Green" title="E28">A6</font></i></a>, <a class="ref" href="sppol_1.html#D1" title="SPPOL_1:def.1">SPPOL_1:def 1</a></i>;<br/>


</div>
