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

<b>let </b><font color="Maroon">p1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">p2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">q1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">q2</font> be   <a href="pre_topc.html#NM2">Point</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="E1:53"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">q1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; <font color="Maroon">q2</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; <font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:53"/><i><font color="Green">E51</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="sppol_1.html#T21">SPPOL_1:21</a></i>;<br/>
<b>consider </b><font color="Maroon">r2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:53"/><i><font color="Green">E52</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="sppol_1.html#T21">SPPOL_1:21</a></i>;<br/>
<div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1_1"/>
( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> or <font color="Maroon">r1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1"><b>case </b></a><a NAME="E1:53_1_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font>
 ;<br/><div class="add"><a NAME="E2:53_1_1_1"/>
for <font color="Olive">s1</font>, <font color="Olive">s2</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s1</font> &amp; <font color="Olive">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">s1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Olive">s1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s2</font> &amp; <font color="Olive">s2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">s2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Olive">s2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> holds <br/><font color="Olive">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s1</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">s2</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:53_1_1_1_1"/><i><font color="Green">E62</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">s2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> )
 ;<br/>

<a NAME="E2:53_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan3.html#T18" target="_self">Th18</a>, <a class="ref" href="euclid.html#T49">EUCLID:49</a></i>;<br/>
<a NAME="E3:53_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T49">EUCLID:49</a></i>;<br/>
<a NAME="E4:53_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T46">EUCLID:46</a></i>;<br/>
<a NAME="E5:53_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>;<br/>
<a NAME="E6:53_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T54">EUCLID:54</a></i>;<br/>
<a NAME="E7:53_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T49">EUCLID:49</a></i>;<br/>
<a NAME="E8:53_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T46">EUCLID:46</a></i>;<br/>
<a NAME="E9:53_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>;<br/>
<a NAME="E10:53_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T54">EUCLID:54</a></i>;<br/>
<a NAME="E11:53_1_1_1_1"/><b>then </b>
( <font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 
<b>by </b><i><a class="ref" href="euclid.html#T38">EUCLID:38</a></i>;<br/>
<a NAME="E12:53_1_1_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E13:53_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan3.html#T19" target="_self">Th19</a>, , <a class="ref" href="euclid.html#T49">EUCLID:49</a></i>;<br/>
<a NAME="E14:53_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T49">EUCLID:49</a></i>;<br/>
<a NAME="E15:53_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T46">EUCLID:46</a></i>;<br/>
<a NAME="E16:53_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>;<br/>
<a NAME="E17:53_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T54">EUCLID:54</a></i>;<br/>
<a NAME="E18:53_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T49">EUCLID:49</a></i>;<br/>
<a NAME="E19:53_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T46">EUCLID:46</a></i>;<br/>
<a NAME="E20:53_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>;<br/>
<a NAME="E21:53_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T54">EUCLID:54</a></i>;<br/>
<a NAME="E22:53_1_1_1_1"/><b>then </b>
( <font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 
<b>by </b><i><a class="ref" href="euclid.html#T38">EUCLID:38</a></i>;<br/>
<b>hence </b><a NAME="E23:53_1_1_1_1"/>
<font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font>
 <b>by </b><i>, , <a class="txt" href="jordan3.html#E1:53_1_1_1_1"><i><font color="Green">E62</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:53_1_1_1"/>
(  <a href="jordan3.html#R2" target="_self">LE</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> or  <a href="jordan3.html#R3" target="_self">LT</a> <font color="Maroon">q2</font>,<font color="Maroon">q1</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 <b>by </b><i>, <a class="txt" href="jordan3.html#E6:53"><i><font color="Green">E53</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_2"><b>case </b></a><a NAME="E1:53_1_1_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r2</font>
 ;<br/><div class="add"><a NAME="E2:53_1_1_2"/>
for <font color="Olive">s2</font>, <font color="Olive">s1</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s2</font> &amp; <font color="Olive">s2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">s2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Olive">s2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s1</font> &amp; <font color="Olive">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">s1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Olive">s1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> holds <br/><font color="Olive">s1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">s2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s2</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">s1</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:53_1_1_2_1"/><i><font color="Green">E65</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s2</font> &amp; <font color="Maroon">s2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">s1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> )
 ;<br/>

<a NAME="E2:53_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan3.html#T18" target="_self">Th18</a>, <a class="ref" href="euclid.html#T49">EUCLID:49</a></i>;<br/>
<a NAME="E3:53_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T49">EUCLID:49</a></i>;<br/>
<a NAME="E4:53_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T46">EUCLID:46</a></i>;<br/>
<a NAME="E5:53_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>;<br/>
<a NAME="E6:53_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T54">EUCLID:54</a></i>;<br/>
<a NAME="E7:53_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T49">EUCLID:49</a></i>;<br/>
<a NAME="E8:53_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T46">EUCLID:46</a></i>;<br/>
<a NAME="E9:53_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>;<br/>
<a NAME="E10:53_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T54">EUCLID:54</a></i>;<br/>
<a NAME="E11:53_1_1_2_1"/><b>then </b>
( <font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 
<b>by </b><i><a class="ref" href="euclid.html#T38">EUCLID:38</a></i>;<br/>
<a NAME="E12:53_1_1_2_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E13:53_1_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><font color="Maroon">s2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan3.html#T19" target="_self">Th19</a>, , <a class="ref" href="euclid.html#T49">EUCLID:49</a></i>;<br/>
<a NAME="E14:53_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T49">EUCLID:49</a></i>;<br/>
<a NAME="E15:53_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T46">EUCLID:46</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><font color="Maroon">s2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T54">EUCLID:54</a></i>
;<br/>
<a NAME="E17:53_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T49">EUCLID:49</a></i>;<br/>
<a NAME="E18:53_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#T46">EUCLID:46</a></i>;<br/>
<a NAME="E19:53_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>;<br/>
<a NAME="E20:53_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T54">EUCLID:54</a></i>;<br/>
<a NAME="E21:53_1_1_2_1"/><b>then </b>
( <font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 
<b>by </b><i><a class="ref" href="euclid.html#T38">EUCLID:38</a></i>;<br/>
<b>hence </b><a NAME="E22:53_1_1_2_1"/>
<font color="Maroon">s1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">s2</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div><a NAME="E3:53_1_1_2"/><b>then </b>
(  <a href="jordan3.html#R2" target="_self">LE</a> <font color="Maroon">q2</font>,<font color="Maroon">q1</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; <font color="Maroon">q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q2</font> )
 <b>by </b><i>, <a class="ref" href="jordan3.html#T18" target="_self">Th18</a>, <a class="ref" href="jordan3.html#T19" target="_self">Th19</a>, , <a class="txt" href="jordan3.html#E6:53"><i><font color="Green">E53</font></i></a></i>;<br/><b>hence </b><a NAME="E4:53_1_1_2"/>
(  <a href="jordan3.html#R2" target="_self">LE</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> or  <a href="jordan3.html#R3" target="_self">LT</a> <font color="Maroon">q2</font>,<font color="Maroon">q1</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_2"/><i><font color="Green">E67</font></i>: 
(  <a href="jordan3.html#R2" target="_self">LE</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp;  <a href="jordan3.html#R3" target="_self">LT</a> <font color="Maroon">q2</font>,<font color="Maroon">q1</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/><a NAME="E2:53_2"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font>
 <b>by </b><i><a class="ref" href="jordan3.html#T18" target="_self">Th18</a>, <a class="ref" href="jordan3.html#T19" target="_self">Th19</a>, <a class="txt" href="jordan3.html#E6:53"><i><font color="Green">E53</font></i></a></i>;<br/><a NAME="E3:53_2"/>
(  <a href="jordan3.html#R2" target="_self">LE</a> <font color="Maroon">q2</font>,<font color="Maroon">q1</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; <font color="Maroon">q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q2</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E4:53_2"/><b>then </b>
<font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font>
 <b>by </b><i><a class="ref" href="jordan3.html#T18" target="_self">Th18</a>, <a class="ref" href="jordan3.html#T19" target="_self">Th19</a>, <a class="txt" href="jordan3.html#E6:53"><i><font color="Green">E53</font></i></a></i>;<br/><a NAME="E5:53_2"/><b>then </b>
<font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E6:53_2"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan3.html#T18" target="_self">Th18</a>, <a class="ref" href="jordan3.html#T19" target="_self">Th19</a>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:53"/>
( (  <a href="jordan3.html#R2" target="_self">LE</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> or  <a href="jordan3.html#R3" target="_self">LT</a> <font color="Maroon">q2</font>,<font color="Maroon">q1</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> ) &amp; (  not  <a href="jordan3.html#R2" target="_self">LE</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> or  not  <a href="jordan3.html#R3" target="_self">LT</a> <font color="Maroon">q2</font>,<font color="Maroon">q1</font>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font> ) )
 <b>by </b><i><a class="ref" href="jordan3.html#T21" target="_self">Th21</a></i>;<br/>


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