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

<b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">b</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">c</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">d</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>set </b><font color="Maroon">K</font> =  <a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>;<br/>
<b>assume </b><a NAME="E1:76"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">d</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">a2</font> = <font color="Maroon">a</font>, <font color="Maroon">b2</font> = <font color="Maroon">b</font>, <font color="Maroon">c2</font> = <font color="Maroon">c</font>, <font color="Maroon">d2</font> = <font color="Maroon">d</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> holds <br/>( ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> implies <font color="Maroon">a<sub>2</sub></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> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Olive">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Olive">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> ) &amp; ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> implies <font color="Maroon">a<sub>2</sub></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Olive">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="real_1.html#K4">*</a> <font color="Olive">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> ) );<br/>
<a NAME="E3:76"/><i><font color="Green">E30</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="heine.html#T2">HEINE:2</a></i>;<br/>
<a NAME="E4:76"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y2</font>] holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:76_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y2</font>] )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> )
 </span><b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_1_1"><b>case </b></a><a NAME="E1:76_1_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">r</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> ;<br/><a NAME="E3:76_1_1_1_1"/>
<font color="Maroon">y1</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> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:76_1_1_1_1"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_1_1_1_2"><b>case </b></a><a NAME="E1:76_1_1_1_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">r</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> ;<br/><a NAME="E3:76_1_1_1_2"/>
<font color="Maroon">y1</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:76_1_1_1_2"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:76_1"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:76"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:76_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_2_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> )
 </span><b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_2_1_1_1"><b>case </b></a><a NAME="E1:76_2_1_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">r</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> ;<br/><a NAME="E3:76_2_1_1_1"/><i><font color="Green">E78</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 )
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><b>set </b><font color="Maroon">y0</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>;<br/><a NAME="E4:76_2_1_1_1"/>
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:76_2_1_1_1_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<a NAME="E2:76_2_1_1_1_1"/><b>then </b>
( 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E3:76_2_1_1_1_1"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<b>then </b><i><font color="Green">E92</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> = 
<span class="p1">(<span class="default">0 <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span> <a href="euclid.html#K17">+</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></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">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> = 
<span class="p1">(<span class="default">1 <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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#T33">EUCLID:33</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></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#T33">EUCLID:33</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>
;<br/>
<b>hence </b><a NAME="E6:76_2_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E5:76_2_1_1_1"/><b>then </b>
for <font color="Olive">r2</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">r2</font> holds <br/>( ( <font color="Olive">r2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></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> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Olive">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Olive">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> ) &amp; ( <font color="Olive">r2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Olive">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="real_1.html#K4">*</a> <font color="Olive">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> ) )
 ;<br/><b>hence </b><a NAME="E6:76_2_1_1_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_2_1_1_2"><b>case </b></a><a NAME="E1:76_2_1_1_2"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">r</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> ;<br/><a NAME="E3:76_2_1_1_2"/><i><font color="Green">E96</font></i>: 
( 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><b>set </b><font color="Maroon">y0</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>;<br/><a NAME="E4:76_2_1_1_2"/>
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></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> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:76_2_1_1_2_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<a NAME="E2:76_2_1_1_2_1"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 )
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E3:76_2_1_1_2_1"/><b>then </b><i><font color="Green">E98</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<b>then </b><i><font color="Green">E99</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default">0 <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></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#T33">EUCLID:33</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></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">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default">1 <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span> <a href="euclid.html#K17">+</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>
;<br/>
<b>hence </b><a NAME="E6:76_2_1_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></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> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E5:76_2_1_1_2"/><b>then </b>
for <font color="Olive">r2</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">r2</font> holds <br/>( ( <font color="Olive">r2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></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> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Olive">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Olive">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> ) &amp; ( <font color="Olive">r2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Olive">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="real_1.html#K4">*</a> <font color="Olive">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> ) )
 ;<br/><b>hence </b><a NAME="E6:76_2_1_1_2"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:76_2"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:76"/>
 ex <font color="Olive">f2</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>(  <a href="relat_1.html#K1">dom</a> <font color="Olive">f2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>] ) )
 
<b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(, );<br/></i>
<b>then consider </b><font color="Maroon">f2</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:76"/><i><font color="Green">E100</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>] ) )
 ;<br/>
<a NAME="E9:76"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:76_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:76_3"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_3_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> )
 </span><b>by </b><i>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_3_1_1_1"><b>case </b></a><a NAME="E1:76_3_1_1_1"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">r</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> ;<br/><a NAME="E3:76_3_1_1_1"/><i><font color="Green">E103</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 )
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E4:76_3_1_1_1"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">r</font> <a href="real_1.html#K4">*</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E5:76_3_1_1_1"/><i><font color="Green">E124</font></i>: 
2 <a href="nat_1.html#K2">*</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E6:76_3_1_1_1"/>
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</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> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E7:76_3_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">lambda</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Olive">lambda</font> <a href="euclid.html#K18">*</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> where <font color="Olive">lambda</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">lambda</font> &amp; <font color="Olive">lambda</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i>, , </i>;<br/><a NAME="E8:76_3_1_1_1"/><b>then </b><i><font color="Green">E125</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>
 ;<br/><a NAME="E9:76_3_1_1_1"/>
 <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E10:76_3_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E11:76_3_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_3_1_1_2"><b>case </b></a><a NAME="E1:76_3_1_1_2"/><i><font color="Green">E158</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">r</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> ;<br/><a NAME="E3:76_3_1_1_2"/><i><font color="Green">E159</font></i>: 
( 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E4:76_3_1_1_2"/><b>then </b>
<font color="Maroon">r</font> <a href="real_1.html#K4">*</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E5:76_3_1_1_2"/><b>then </b><i><font color="Green">E162</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E6:76_3_1_1_2"/>
2 <a href="nat_1.html#K2">*</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E7:76_3_1_1_2"/><b>then </b><i><font color="Green">E163</font></i>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E8:76_3_1_1_2"/>
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E9:76_3_1_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">lambda</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Olive">lambda</font> <a href="euclid.html#K18">*</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> where <font color="Olive">lambda</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">lambda</font> &amp; <font color="Olive">lambda</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i>, , </i>;<br/><a NAME="E10:76_3_1_1_2"/><b>then </b><i><font color="Green">E164</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>
 ;<br/><a NAME="E11:76_3_1_1_2"/>
 <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E12:76_3_1_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E13:76_3_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:76_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f3</font> = <font color="Maroon">f2</font> as   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span></span>)</span> <b>by </b><i>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E11:76"/><i><font color="Green">E165</font></i>: 
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E12:76"/>
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<b>then </b><i><font color="Green">E166</font></i>: <font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> 0 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></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#T33">EUCLID:33</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></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#T33">EUCLID:33</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>
<br/>.= 
 <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
<b>by </b><i>, </i>
;<br/>
<a NAME="E14:76"/><i><font color="Green">E167</font></i>: 
1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E15:76"/>
1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<b>then </b><i><font color="Green">E169</font></i>: <font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default">0 <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span> <a href="euclid.html#K17">+</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>
<br/>.= 
 <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
<b>by </b><i>, </i>
;<br/>
<a NAME="E17:76"/><i><font color="Green">E171</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> holds <br/><font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</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> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Olive">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Olive">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_6"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:76_6"/><i><font color="Green">E172</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<a NAME="E2:76_6"/><b>then </b><i><font color="Green">E173</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2 )
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E3:76_6"/><b>then </b>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:76_6"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<b>hence </b><a NAME="E5:76_6"/>
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</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> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:76"/><i><font color="Green">E175</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> holds <br/><font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Olive">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="real_1.html#K4">*</a> <font color="Olive">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_7"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:76_7"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<a NAME="E2:76_7"/><b>then </b><i><font color="Green">E179</font></i>: 
( 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E3:76_7"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<b>hence </b><a NAME="E4:76_7"/>
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:76"/><i><font color="Green">E180</font></i>: 
for <font color="Olive">p</font> being  <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>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> holds <br/>( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</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="E1:76_8"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>
 ;<br/>

<a NAME="E2:76_8"/><b>then </b><i><font color="Green">E182</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">c2</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">d2</font></span><a href="euclid.html#K23">]|</a></span>
 
;<br/>
<a NAME="E3:76_8"/><i><font color="Green">E184</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E4:76_8"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E5:76_8"/><b>then </b><i><font color="Green">E206</font></i>: 
( <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> )
 
<b>by </b><i>, , <a class="ref" href="jgraph_6.html#T15" target="_self">Th15</a>, <a class="ref" href="topreal1.html#T10">TOPREAL1:10</a></i>;<br/>
<a NAME="E6:76_8"/><i><font color="Green">E207</font></i>: 
<font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>
<a NAME="E7:76_8"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E8:76_8"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E9:76_8"/><b>then </b><i><font color="Green">E208</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E10:76_8"/><i><font color="Green">E209</font></i>: 
<font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>
<a NAME="E11:76_8"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E12:76_8"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E13:76_8"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/>
<a NAME="E14:76_8"/><b>then </b><i><font color="Green">E210</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<b>set </b><font color="Maroon">r</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2;<br/>
<a NAME="E15:76_8"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="jgraph_6.html#T16" target="_self">Th16</a>, <a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<b>then </b><font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p0">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T62">EUCLID:62</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K17">+</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p0">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p0">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">d</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T62">EUCLID:62</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">d</font></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T60">EUCLID:60</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>

<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i>, <a class="ref" href="topreal3.html#T17">TOPREAL3:17</a></i>
<br/>.= 
<font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>
;<br/>
<b>hence </b><a NAME="E17:76_8"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="jgraph_6.html#T16" target="_self">Th16</a>, <a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:76"/><i><font color="Green">E211</font></i>: 
for <font color="Olive">p</font> being  <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>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> holds <br/>( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</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="E1:76_9"/><i><font color="Green">E212</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>
 ;<br/>

<a NAME="E2:76_9"/><b>then </b><i><font color="Green">E213</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">d2</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b2</font>,<font color="Maroon">d2</font></span><a href="euclid.html#K23">]|</a></span>
 
;<br/>
<a NAME="E3:76_9"/><i><font color="Green">E214</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E4:76_9"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E5:76_9"/><b>then </b><i><font color="Green">E215</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, , , <a class="ref" href="topreal1.html#T9">TOPREAL1:9</a></i>;<br/>
<a NAME="E6:76_9"/><i><font color="Green">E216</font></i>: 
<font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>
<a NAME="E7:76_9"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E8:76_9"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jgraph_6.html#T16" target="_self">Th16</a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E9:76_9"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E10:76_9"/><b>then </b><i><font color="Green">E217</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E11:76_9"/><i><font color="Green">E218</font></i>: 
<font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>
<a NAME="E12:76_9"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E13:76_9"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E14:76_9"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/>
<a NAME="E15:76_9"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E16:76_9"/><b>then </b><i><font color="Green">E219</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>set </b><font color="Maroon">r</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>;<br/>
<a NAME="E17:76_9"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, , <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<b>then </b><font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">d</font></span>)</span></span><a href="euclid.html#K23">]|</a></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">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T62">EUCLID:62</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">d</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K17">+</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">d</font></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T62">EUCLID:62</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">d</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">d</font></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T60">EUCLID:60</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>

<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="jgraph_6.html#T15" target="_self">Th15</a>, <a class="ref" href="topreal3.html#T18">TOPREAL3:18</a></i>
<br/>.= 
<font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>
;<br/>
<b>hence </b><a NAME="E19:76_9"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">B00</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B01</font> = <font color="Maroon">B00</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E23:76"/>
 <a href="topmetr.html#K5">I[01]</a>  <a href="hidden.html#R1">=</a> <a href="topmetr.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B01</font>
 
<b>by </b><i><a class="ref" href="topmetr.html#T26">TOPMETR:26</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<b>then consider </b><font color="Maroon">h1</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E25:76"/><i><font color="Green">E220</font></i>: 
( ( for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  holds  <font color="Maroon">h1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> ) &amp; <font color="Maroon">h1</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">h2</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E27:76"/><i><font color="Green">E221</font></i>: 
( ( for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <br/> for <font color="Olive">r1</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">h1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> holds <br/><font color="Maroon">h2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> 2 <a href="xcmplx_0.html#K3">*</a> <font color="Olive">r1</font> ) &amp; <font color="Maroon">h2</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i>, <a class="ref" href="jgraph_2.html#T33">JGRAPH_2:33</a></i>;<br/>
<b>consider </b><font color="Maroon">h5</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E29:76"/><i><font color="Green">E222</font></i>: 
( ( for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <br/> for <font color="Olive">r1</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">h2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> holds <br/><font color="Maroon">h5</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K6">-</a> <font color="Olive">r1</font> ) &amp; <font color="Maroon">h5</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i><a class="ref" href="jgraph_6.html#T15" target="_self">Th15</a>, </i>;<br/>
<b>consider </b><font color="Maroon">h3</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E31:76"/><i><font color="Green">E223</font></i>: 
( ( for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <br/> for <font color="Olive">r1</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">h2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> holds <br/><font color="Maroon">h3</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> <a href="xcmplx_0.html#K6">-</a> 1 ) &amp; <font color="Maroon">h3</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i><a class="ref" href="jgraph_6.html#T15" target="_self">Th15</a>, </i>;<br/>
<b>consider </b><font color="Maroon">h4</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E33:76"/><i><font color="Green">E224</font></i>: 
( ( for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <br/> for <font color="Olive">r1</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">h3</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">r1</font> holds <br/><font color="Maroon">h4</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K6">-</a> <font color="Olive">r1</font> ) &amp; <font color="Maroon">h4</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">g1</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E35:76"/><i><font color="Green">E225</font></i>: 
( ( for <font color="Olive">r</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  holds  <font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h5</font> <a href="funct_2.html#K3">.</a> <font color="Olive">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> ) &amp; <font color="Maroon">g1</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i><a class="ref" href="jgraph_6.html#T15" target="_self">Th15</a>, , <a class="ref" href="jgraph_6.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E36:76"/><i><font color="Green">E226</font></i>: 
for <font color="Olive">r</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <br/> for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> holds <br/><font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Olive">s</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Olive">s</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/><b>let </b><font color="Maroon">s</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>assume </b><a NAME="E1:76_10"/><i><font color="Green">E227</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 ;<br/>

<font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h5</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">h2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">h1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="jgraph_6.html#T15" target="_self">Th15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">h1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="jgraph_6.html#T15" target="_self">Th15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">h1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
<b>by </b><i>, </i>
;<br/>
<b>hence </b><a NAME="E3:76_10"/>
<font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">g2</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E38:76"/><i><font color="Green">E228</font></i>: 
( ( for <font color="Olive">r</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  holds  <font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h4</font> <a href="funct_2.html#K3">.</a> <font color="Olive">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h3</font> <a href="funct_2.html#K3">.</a> <font color="Olive">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> ) &amp; <font color="Maroon">g2</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i>, <a class="ref" href="jgraph_6.html#T16" target="_self">Th16</a>, <a class="ref" href="jgraph_6.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E39:76"/><i><font color="Green">E229</font></i>: 
for <font color="Olive">r</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <br/> for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> holds <br/><font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Olive">s</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="xcmplx_0.html#K3">*</a> <font color="Olive">s</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/><b>let </b><font color="Maroon">s</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>assume </b><a NAME="E1:76_11"/><i><font color="Green">E230</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 ;<br/>

<font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h4</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h3</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">h3</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h3</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="jgraph_6.html#T16" target="_self">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">h2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h3</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">h2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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"><font color="Maroon">h2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">h1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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"><font color="Maroon">h2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="jgraph_6.html#T15" target="_self">Th15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">h1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">h1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="jgraph_6.html#T15" target="_self">Th15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">h1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
<b>by </b><i>, </i>
;<br/>
<b>hence </b><a NAME="E3:76_11"/>
<font color="Maroon">g2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">B11</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<i><font color="Green">E231</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B11</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B11</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  <a href="subset_1.html#K9">/\</a> <font color="Maroon">B11</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<font color="Maroon">B11</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B11</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>
;<br/>
<a NAME="E42:76"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B11</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
;<br/>
<b>then reconsider </b><font color="Maroon">g11</font> = <font color="Maroon">g1</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B11</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B11</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E44:76"/><i><font color="Green">E232</font></i>: 
 <a href="euclid.html#K15">TOP-REAL</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T3">TSEP_1:3</a></i>;<br/>
<a NAME="E45:76"/><b>then </b><i><font color="Green">E233</font></i>: 
<font color="Maroon">g11</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, <a class="ref" href="borsuk_4.html#T69">BORSUK_4:69</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B22</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<i><font color="Green">E234</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B22</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B22</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  <a href="subset_1.html#K9">/\</a> <font color="Maroon">B22</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<font color="Maroon">B22</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B22</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>
;<br/>
<a NAME="E48:76"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B22</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
;<br/>
<b>then reconsider </b><font color="Maroon">g22</font> = <font color="Maroon">g2</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B22</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B22</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E50:76"/><i><font color="Green">E235</font></i>: 
<font color="Maroon">g22</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, , <a class="ref" href="borsuk_4.html#T69">BORSUK_4:69</a></i>;<br/>
<a NAME="E51:76"/><i><font color="Green">E236</font></i>: 
<font color="Maroon">B11</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B11</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E52:76"/><i><font color="Green">E237</font></i>: 
<font color="Maroon">B22</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B22</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E53:76"/><i><font color="Green">E238</font></i>: 
<font color="Maroon">B11</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i/>;<br/>
<a NAME="E54:76"/><i><font color="Green">E239</font></i>: 
<font color="Maroon">B22</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i/>;<br/>
<a NAME="E55:76"/>
<font color="Maroon">B11</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B22</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="heine.html#T2">HEINE:2</a></i>;<br/>
<a NAME="E56:76"/><b>then </b><i><font color="Green">E240</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B11</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B22</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i>, , <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E57:76"/>
for <font color="Olive">p</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B11</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B22</font></span>)</span></span>)</span> holds <br/><font color="Maroon">g11</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g22</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_14"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:76_14"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B11</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B22</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:76_14"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B11</font></span>)</span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B22</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:76_14"/><b>then </b><i><font color="Green">E298</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B11</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B22</font> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">rp</font> = <font color="Maroon">p</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E5:76_14"/><i><font color="Green">E300</font></i>: 
<font color="Maroon">rp</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E6:76_14"/>
<font color="Maroon">rp</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E7:76_14"/><b>then </b>
<font color="Maroon">rp</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E8:76"><i><font color="Green">E100</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E8:76_14"/><b>then </b><i><font color="Green">E301</font></i>: 
2 <a href="real_1.html#K4">*</a> <font color="Maroon">rp</font> <a href="hidden.html#R1">=</a> 1
 
;<br/>
<b>thus </b><font color="Maroon">g11</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> = 
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default">1 <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default">1 <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">g22</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">h</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E59:76"/><i><font color="Green">E302</font></i>: 
( <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g11</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g22</font> &amp; <font color="Maroon">h</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i>, , , , , , , <a class="ref" href="jgraph_2.html#T9">JGRAPH_2:9</a></i>;<br/>
<a NAME="E60:76"/><i><font color="Green">E303</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f3</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f3</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E61:76"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> holds <br/><font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_15"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:76_15"/><i><font color="Green">E304</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">rx</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="jgraph_6.html#E8:76"><i><font color="Green">E100</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/>
<a NAME="E3:76_15"/><i><font color="Green">E306</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">rx</font> &amp; <font color="Maroon">rx</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E8:76"><i><font color="Green">E100</font></i></a>, , <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_15_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_15_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_15_1_1"/>
( <font color="Maroon">rx</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 or <font color="Maroon">rx</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K6">/</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_15_1_1_1"><b>case </b></a><a NAME="E1:76_15_1_1_1"/><i><font color="Green">E307</font></i>: 
<font color="Maroon">rx</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2
 ;<br/><div class="add"><a NAME="E2:76_15_1_1_1"/><b>then </b><i><font color="Green">E308</font></i>: 
<font color="Maroon">rx</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_15_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:76_15_1_1_1_1"/>
<font color="Maroon">rx</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g22</font>
 ;<br/><a NAME="E2:76_15_1_1_1_1"/><b>then </b>
<font color="Maroon">rx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B22</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:76_15_1_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/></div><b>end;</b></div><b>then </b><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">rx</font> = 
<font color="Maroon">g11</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">rx</font>
<b>by </b><i>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">rx</font>
<b>by </b><i><a class="ref" href="jgraph_6.html#T18" target="_self">Th18</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">rx</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">rx</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="jgraph_6.html#T17" target="_self">Th17</a>, <a class="txt" href="jgraph_6.html#E8:76"><i><font color="Green">E100</font></i></a>, </i>
<br/>.= 
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">rx</font>
<b>by </b><i>, <a class="ref" href="jgraph_6.html#T18" target="_self">Th18</a></i>
;<br/><b>hence </b><a NAME="E5:76_15_1_1_1"/>
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_15_1_1_2"><b>case </b></a><a NAME="E1:76_15_1_1_2"/>
<font color="Maroon">rx</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K6">/</a> 2
 ;<br/><div class="add"><a NAME="E2:76_15_1_1_2"/><b>then </b><i><font color="Green">E309</font></i>: 
<font color="Maroon">rx</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E3:76_15_1_1_2"/><b>then </b>
<font color="Maroon">rx</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B22</font></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/><b>then </b><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">rx</font> = 
<font color="Maroon">g22</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">rx</font>
<b>by </b><i>, , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">rx</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">rx</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="real_1.html#K4">*</a> <font color="Maroon">rx</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
<b>by </b><i>, <a class="txt" href="jgraph_6.html#E8:76"><i><font color="Green">E100</font></i></a>, </i>
<br/>.= 
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">rx</font>
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E5:76_15_1_1_2"/>
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:76_15"/>
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E62:76"/><b>then </b>
<font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 
<b>by </b><i><a class="txt" href="jgraph_6.html#E8:76"><i><font color="Green">E100</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E63:76"/><b>then </b><i><font color="Green">E310</font></i>: 
<font color="Maroon">f3</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, <a class="ref" href="jgraph_1.html#T63">JGRAPH_1:63</a></i>;<br/>
<a NAME="E64:76"/><i><font color="Green">E311</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f3</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E65:76"/>
for <font color="Olive">x1</font>, <font color="Olive">x2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f3</font> &amp; <font color="Olive">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f3</font> &amp; <font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x2</font> holds <br/><font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Olive">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_16"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:76_16"/><i><font color="Green">E313</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f3</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f3</font> &amp; <font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">r1</font> = <font color="Maroon">x1</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">r2</font> = <font color="Maroon">x2</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, </i>;<br/>
<a NAME="E4:76_16"/><i><font color="Green">E314</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_16_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_16_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_16_1_1"/>
( ( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> ) or ( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> ) or ( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> ) or ( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> ) )
 </span><b>by </b><i>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_16_1_1_1"><b>case </b></a><a NAME="E1:76_16_1_1_1"/><i><font color="Green">E315</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> )
 ;<br/><div class="add"><a NAME="E2:76_16_1_1_1"/><b>then </b>
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</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> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:76_16_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> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></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> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="jgraph_6.html#T20" target="_self">Th20</a></i>;<br/><a NAME="E4:76_16_1_1_1"/><b>then </b>
<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> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="euclid.html#T52">EUCLID:52</a></i>;<br/><a NAME="E5:76_16_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> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="euclid.html#T49">EUCLID:49</a></i>;<br/><a NAME="E6:76_16_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> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="euclid.html#T54">EUCLID:54</a></i>;<br/><a NAME="E7:76_16_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></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> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></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> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></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> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#T49">EUCLID:49</a></i>;<br/><a NAME="E8:76_16_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></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> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="ref" href="euclid.html#T46">EUCLID:46</a></i>;<br/><a NAME="E9:76_16_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="ref" href="euclid.html#T54">EUCLID:54</a></i>;<br/><a NAME="E10:76_16_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 ;<br/><a NAME="E11:76_16_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="ref" href="euclid.html#T44">EUCLID:44</a></i>;<br/><a NAME="E12:76_16_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID:45</a></i>;<br/><a NAME="E13:76_16_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="ref" href="euclid.html#T53">EUCLID:53</a></i>;<br/><a NAME="E14:76_16_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="hidden.html#R1">=</a> 0 or <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 <b>by </b><i><a class="ref" href="euclid.html#T35">EUCLID:35</a></i>;<br/><a NAME="E15:76_16_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="hidden.html#R1">=</a> 0 or <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> )
 <b>by </b><i><a class="ref" href="euclid.html#T47">EUCLID:47</a></i>;<br/><a NAME="E16:76_16_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E17:76_16_1_1_1"/><b>then </b>
2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font>
 <b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E18:76_16_1_1_1"/><b>then </b>
<font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2
 ;<br/><b>hence </b><a NAME="E19:76_16_1_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_16_1_1_2"><b>case </b></a><a NAME="E1:76_16_1_1_2"/><i><font color="Green">E316</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> )
 ;<br/><div class="add"><a NAME="E2:76_16_1_1_2"/><b>then </b><i><font color="Green">E317</font></i>: 
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</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> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:76_16_1_1_2"/><i><font color="Green">E318</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 <a href="real_1.html#K6">/</a> 2 )
 <b>by </b><i><a class="ref" href="jgraph_6.html#T20" target="_self">Th20</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E4:76_16_1_1_2"/><b>then </b><i><font color="Green">E319</font></i>: 
<font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E5:76_16_1_1_2"/>
2 <a href="nat_1.html#K2">*</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E6:76_16_1_1_2"/><b>then </b>
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">lambda</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Olive">lambda</font> <a href="euclid.html#K18">*</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> where <font color="Olive">lambda</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">lambda</font> &amp; <font color="Olive">lambda</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i>, </i>;<br/><a NAME="E7:76_16_1_1_2"/><b>then </b><i><font color="Green">E320</font></i>: 
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>
 ;<br/><a NAME="E8:76_16_1_1_2"/><i><font color="Green">E321</font></i>: 
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r2</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="jgraph_6.html#T20" target="_self">Th20</a></i>;<br/><a NAME="E9:76_16_1_1_2"/><i><font color="Green">E322</font></i>: 
( 1 <a href="real_1.html#K6">/</a> 2 <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 )
 <b>by </b><i><a class="ref" href="jgraph_6.html#T20" target="_self">Th20</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E10:76_16_1_1_2"/><b>then </b>
<font color="Maroon">r2</font> <a href="real_1.html#K4">*</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E11:76_16_1_1_2"/><b>then </b><i><font color="Green">E323</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E12:76_16_1_1_2"/>
2 <a href="nat_1.html#K2">*</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E13:76_16_1_1_2"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E14:76_16_1_1_2"/><b>then </b>
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">lambda</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Olive">lambda</font> <a href="euclid.html#K18">*</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> where <font color="Olive">lambda</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">lambda</font> &amp; <font color="Olive">lambda</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i>, </i>;<br/><a NAME="E15:76_16_1_1_2"/><b>then </b>
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>
 ;<br/><a NAME="E16:76_16_1_1_2"/><b>then </b>
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E17:76_16_1_1_2"/><b>then </b><i><font color="Green">E324</font></i>: 
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T18" target="_self">Th18</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E18:76_16_1_1_2"/><b>then </b>
<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> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i>, <a class="ref" href="euclid.html#T40">EUCLID:40</a></i>;<br/><a NAME="E19:76_16_1_1_2"/><b>then </b>
<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> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="ref" href="euclid.html#T43">EUCLID:43</a></i>;<br/><a NAME="E20:76_16_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="ref" href="euclid.html#T30">EUCLID:30</a></i>;<br/><a NAME="E21:76_16_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="ref" href="euclid.html#T37">EUCLID:37</a></i>;<br/><a NAME="E22:76_16_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 ;<br/><a NAME="E23:76_16_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="ref" href="euclid.html#T44">EUCLID:44</a></i>;<br/><a NAME="E24:76_16_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></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> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID:45</a></i>;<br/><a NAME="E25:76_16_1_1_2"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="ref" href="euclid.html#T53">EUCLID:53</a></i>;<br/><a NAME="E26:76_16_1_1_2"/><b>then </b>
( 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="hidden.html#R1">=</a> 0 or <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 <b>by </b><i><a class="ref" href="euclid.html#T35">EUCLID:35</a></i>;<br/><a NAME="E27:76_16_1_1_2"/><b>then </b>
( 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="hidden.html#R1">=</a> 0 or <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> )
 <b>by </b><i><a class="ref" href="euclid.html#T47">EUCLID:47</a></i>;<br/><a NAME="E28:76_16_1_1_2"/><b>then </b>
( 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E29:76_16_1_1_2"/><b>then </b>
1 <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font>
 <b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E30:76_16_1_1_2"/><b>then </b><i><font color="Green">E325</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> 2
 ;<br/><a NAME="E31:76_16_1_1_2"/>
<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> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i>, , , <a class="ref" href="euclid.html#T40">EUCLID:40</a></i>;<br/><a NAME="E32:76_16_1_1_2"/><b>then </b>
<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> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="ref" href="euclid.html#T43">EUCLID:43</a></i>;<br/><a NAME="E33:76_16_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="ref" href="euclid.html#T30">EUCLID:30</a></i>;<br/><a NAME="E34:76_16_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="ref" href="euclid.html#T37">EUCLID:37</a></i>;<br/><a NAME="E35:76_16_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 ;<br/><a NAME="E36:76_16_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="ref" href="euclid.html#T44">EUCLID:44</a></i>;<br/><a NAME="E37:76_16_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID:45</a></i>;<br/><a NAME="E38:76_16_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="ref" href="euclid.html#T53">EUCLID:53</a></i>;<br/><a NAME="E39:76_16_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> 0 or <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 <b>by </b><i><a class="ref" href="euclid.html#T35">EUCLID:35</a></i>;<br/><a NAME="E40:76_16_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> 0 or <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> )
 <b>by </b><i><a class="ref" href="euclid.html#T47">EUCLID:47</a></i>;<br/><a NAME="E41:76_16_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E42:76_16_1_1_2"/><b>then </b>
1 <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font>
 <b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E43:76_16_1_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_16_1_1_3"><b>case </b></a><a NAME="E1:76_16_1_1_3"/><i><font color="Green">E326</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> )
 ;<br/><div class="add"><a NAME="E2:76_16_1_1_3"/><b>then </b><i><font color="Green">E327</font></i>: 
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r2</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> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:76_16_1_1_3"/><i><font color="Green">E328</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 <a href="real_1.html#K6">/</a> 2 )
 <b>by </b><i><a class="ref" href="jgraph_6.html#T20" target="_self">Th20</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E4:76_16_1_1_3"/><b>then </b><i><font color="Green">E329</font></i>: 
<font color="Maroon">r2</font> <a href="real_1.html#K4">*</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E5:76_16_1_1_3"/>
2 <a href="nat_1.html#K2">*</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E6:76_16_1_1_3"/><b>then </b>
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">lambda</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Olive">lambda</font> <a href="euclid.html#K18">*</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> where <font color="Olive">lambda</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">lambda</font> &amp; <font color="Olive">lambda</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i>, </i>;<br/><a NAME="E7:76_16_1_1_3"/><b>then </b><i><font color="Green">E330</font></i>: 
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>
 ;<br/><a NAME="E8:76_16_1_1_3"/><i><font color="Green">E331</font></i>: 
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="jgraph_6.html#T20" target="_self">Th20</a></i>;<br/><a NAME="E9:76_16_1_1_3"/><i><font color="Green">E332</font></i>: 
( 1 <a href="real_1.html#K6">/</a> 2 <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 )
 <b>by </b><i><a class="ref" href="jgraph_6.html#T20" target="_self">Th20</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E10:76_16_1_1_3"/><b>then </b>
<font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E11:76_16_1_1_3"/><b>then </b><i><font color="Green">E333</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E12:76_16_1_1_3"/>
2 <a href="nat_1.html#K2">*</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E13:76_16_1_1_3"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E14:76_16_1_1_3"/><b>then </b>
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">lambda</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Olive">lambda</font> <a href="euclid.html#K18">*</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> where <font color="Olive">lambda</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">lambda</font> &amp; <font color="Olive">lambda</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i>, </i>;<br/><a NAME="E15:76_16_1_1_3"/><b>then </b>
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>
 ;<br/><a NAME="E16:76_16_1_1_3"/><b>then </b>
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E17:76_16_1_1_3"/><b>then </b><i><font color="Green">E334</font></i>: 
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="jgraph_6.html#T18" target="_self">Th18</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E18:76_16_1_1_3"/><b>then </b>
<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> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i>, <a class="ref" href="euclid.html#T40">EUCLID:40</a></i>;<br/><a NAME="E19:76_16_1_1_3"/><b>then </b>
<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> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="ref" href="euclid.html#T43">EUCLID:43</a></i>;<br/><a NAME="E20:76_16_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="ref" href="euclid.html#T30">EUCLID:30</a></i>;<br/><a NAME="E21:76_16_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="ref" href="euclid.html#T37">EUCLID:37</a></i>;<br/><a NAME="E22:76_16_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 ;<br/><a NAME="E23:76_16_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="ref" href="euclid.html#T44">EUCLID:44</a></i>;<br/><a NAME="E24:76_16_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></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> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID:45</a></i>;<br/><a NAME="E25:76_16_1_1_3"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="ref" href="euclid.html#T53">EUCLID:53</a></i>;<br/><a NAME="E26:76_16_1_1_3"/><b>then </b>
( 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="hidden.html#R1">=</a> 0 or <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 <b>by </b><i><a class="ref" href="euclid.html#T35">EUCLID:35</a></i>;<br/><a NAME="E27:76_16_1_1_3"/><b>then </b>
( 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="hidden.html#R1">=</a> 0 or <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> )
 <b>by </b><i><a class="ref" href="euclid.html#T47">EUCLID:47</a></i>;<br/><a NAME="E28:76_16_1_1_3"/><b>then </b>
( 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a>  )
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E29:76_16_1_1_3"/><b>then </b>
1 <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font>
 <b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E30:76_16_1_1_3"/><b>then </b><i><font color="Green">E335</font></i>: 
<font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> 2
 ;<br/><a NAME="E31:76_16_1_1_3"/>
<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> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i>, , , <a class="ref" href="euclid.html#T40">EUCLID:40</a></i>;<br/><a NAME="E32:76_16_1_1_3"/><b>then </b>
<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> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="ref" href="euclid.html#T43">EUCLID:43</a></i>;<br/><a NAME="E33:76_16_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="ref" href="euclid.html#T30">EUCLID:30</a></i>;<br/><a NAME="E34:76_16_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="ref" href="euclid.html#T37">EUCLID:37</a></i>;<br/><a NAME="E35:76_16_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 ;<br/><a NAME="E36:76_16_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="ref" href="euclid.html#T44">EUCLID:44</a></i>;<br/><a NAME="E37:76_16_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID:45</a></i>;<br/><a NAME="E38:76_16_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="ref" href="euclid.html#T53">EUCLID:53</a></i>;<br/><a NAME="E39:76_16_1_1_3"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> 0 or <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 <b>by </b><i><a class="ref" href="euclid.html#T35">EUCLID:35</a></i>;<br/><a NAME="E40:76_16_1_1_3"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> 0 or <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> )
 <b>by </b><i><a class="ref" href="euclid.html#T47">EUCLID:47</a></i>;<br/><a NAME="E41:76_16_1_1_3"/><b>then </b>
( <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E42:76_16_1_1_3"/><b>then </b>
1 <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font>
 <b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><b>hence </b><a NAME="E43:76_16_1_1_3"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_16_1_1_4"><b>case </b></a><a NAME="E1:76_16_1_1_4"/><i><font color="Green">E336</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> )
 ;<br/><div class="add"><a NAME="E2:76_16_1_1_4"/><b>then </b>
<font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:76_16_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="jgraph_6.html#T20" target="_self">Th20</a></i>;<br/><a NAME="E4:76_16_1_1_4"/><b>then </b>
<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> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="euclid.html#T52">EUCLID:52</a></i>;<br/><a NAME="E5:76_16_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="euclid.html#T49">EUCLID:49</a></i>;<br/><a NAME="E6:76_16_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="euclid.html#T54">EUCLID:54</a></i>;<br/><a NAME="E7:76_16_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#T49">EUCLID:49</a></i>;<br/><a NAME="E8:76_16_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="ref" href="euclid.html#T46">EUCLID:46</a></i>;<br/><a NAME="E9:76_16_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="ref" href="euclid.html#T54">EUCLID:54</a></i>;<br/><a NAME="E10:76_16_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 ;<br/><a NAME="E11:76_16_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="ref" href="euclid.html#T44">EUCLID:44</a></i>;<br/><a NAME="E12:76_16_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID:45</a></i>;<br/><a NAME="E13:76_16_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2
 <b>by </b><i><a class="ref" href="euclid.html#T53">EUCLID:53</a></i>;<br/><a NAME="E14:76_16_1_1_4"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0 or <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 <b>by </b><i><a class="ref" href="euclid.html#T35">EUCLID:35</a></i>;<br/><a NAME="E15:76_16_1_1_4"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0 or <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> )
 <b>by </b><i><a class="ref" href="euclid.html#T47">EUCLID:47</a></i>;<br/><a NAME="E16:76_16_1_1_4"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a>  )
 <b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E17:76_16_1_1_4"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E18:76_16_1_1_4"/><b>then </b>
2 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1
 ;<br/><a NAME="E19:76_16_1_1_4"/><b>then </b>
<font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> 2</span>)</span> <a href="real_1.html#K6">/</a> 2
 ;<br/><b>hence </b><a NAME="E20:76_16_1_1_4"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:76_16"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E66:76"/><b>then </b><i><font color="Green">E337</font></i>: 
<font color="Maroon">f3</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E67:76"/><i><font color="Green">E338</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E68:76"/>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_17"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:76_17"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:76_17"/><b>then </b><i><font color="Green">E339</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">y</font> as   <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/>
<a NAME="E4:76_17"/><i><font color="Green">E340</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_17_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_17_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_17_1_1"/>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> or <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> )
 </span><b>by </b><i><a class="ref" href="jgraph_6.html#T20" target="_self">Th20</a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_17_1_1_1"><b>case </b></a><a NAME="E1:76_17_1_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:76_17_1_1_1"/><b>then </b><i><font color="Green">E341</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> )
 <b>by </b><i/>;<br/><a NAME="E3:76_17_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><b>hence </b><a NAME="E4:76_17_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f3</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_17_1_1_2"><b>case </b></a><a NAME="E1:76_17_1_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>
 ;<br/><div class="add"><a NAME="E2:76_17_1_1_2"/><b>then </b><i><font color="Green">E342</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> )
 <b>by </b><i/>;<br/><a NAME="E3:76_17_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><b>hence </b><a NAME="E4:76_17_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f3</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:76_17"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f3</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E69:76"/><b>then </b><i><font color="Green">E343</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f3</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jgraph_6.html#T18" target="_self">Th18</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E70:76"/><i><font color="Green">E344</font></i>: 
 <a href="topmetr.html#K5">I[01]</a>  is <a href="compts_1.html#V2">compact</a>
 
<b>by </b><i><a class="ref" href="heine.html#T11">HEINE:11</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<a NAME="E71:76"/>
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span> <a href="compts_1.html#NR2" target="_self">is_T2</a> 
 
<b>by </b><i><a class="ref" href="topmetr.html#T3">TOPMETR:3</a></i>;<br/>
<a NAME="E72:76"/><b>then </b><i><font color="Green">E345</font></i>: 
<font color="Maroon">f3</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i>, , , <a class="ref" href="jgraph_6.html#T20" target="_self">Th20</a>, , <a class="ref" href="compts_1.html#T26">COMPTS_1:26</a></i>;<br/>
<a NAME="E73:76"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f3</font> <a href="hidden.html#R1">=</a>  <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jgraph_6.html#T20" target="_self">Th20</a>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>hence </b><a NAME="E74:76"/>
 ex <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <span class="p3">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span></span>)</span></span>)</span> st <br/>( <font color="Olive">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a>  <a href="jordan6.html#K8">Upper_Arc</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#NK1" target="_self">rectangle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font></span>)</span> &amp; ( for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> holds <br/><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</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> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Olive">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Olive">r</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> ) &amp; ( for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> holds <br/><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Olive">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></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">2 <a href="real_1.html#K4">*</a> <font color="Olive">r</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span></span>)</span> ) &amp; ( for <font color="Olive">p</font> being  <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>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> holds <br/>( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">p</font> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> ) ) &amp; ( for <font color="Olive">p</font> being  <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>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">d</font></span><a href="euclid.html#K23">]|</a></span> holds <br/>( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive">p</font> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> ) ) )
 <b>by </b><i>, , , , , , </i>;<br/>


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