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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c4">p2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1"/><i><font color="Green" title="E12">A1</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>
 ;<br/><a NAME="E2:15_1"/><b>then </b>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">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" title="REAL_1:func.5">-</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">p2</font></span>)</span></span>)</span> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c5">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:15_1"/><i><font color="Green" title="E13">A2</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">p2</font></span>)</span>
 <b>and </b><br/><a NAME="E5:15_1"/><i><font color="Green" title="E14">A3</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font>
 <b>and </b><br/><a NAME="E6:15_1"/><i><font color="Green" title="E15">A4</font></i>: 
<font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1_1_1"/>
( ( 0 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon" title="c5">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 ) or  not 0 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">r</font> or  not <font color="Maroon" title="c5">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_1"><b>suppose </b></a><a NAME="E1:15_1_1_1_1"/><i><font color="Green" title="E16">A5</font></i>: 
( 0 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon" title="c5">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 ;<br/><div class="add"><a NAME="E2:15_1_1_1_1"/>
(  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> )
 <b>by </b><i><a class="txt" href="topreal1.html#E1:15_1"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="topreal1.html#E15"><i><font color="Green" title="E10">Lm3</font></i></a></i>;<br/><a NAME="E3:15_1_1_1_1"/><b>then </b><i><font color="Green" title="E17">A6</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><a NAME="E4:15_1_1_1_1"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:15_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>
 ;<br/><a NAME="E2:15_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">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" title="REAL_1:func.5">-</a> <font color="Olive" title="b1">lambda</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">lambda</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">p2</font></span>)</span></span>)</span> where <font color="Olive" title="b1">lambda</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">lambda</font> &amp; <font color="Olive" title="b1">lambda</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c7">b</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:15_1_1_1_1_1_1"/><i><font color="Green" title="E18">A7</font></i>: 
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">p2</font></span>)</span>
 <b>and </b><br/><a NAME="E5:15_1_1_1_1_1_1"/><i><font color="Green" title="E19">A8</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c7">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c7">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> or  not <font color="Maroon" title="c7">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:15_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E20">A9</font></i>: 
<font color="Maroon" title="c7">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font>
 ;<br/><div class="add"><a NAME="E2:15_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E21">A10</font></i>: 
1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="topreal1.html#E5:15_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="real_2.html#T125" title="REAL_2:th.125">REAL_2:125</a></i>;<br/><b>set </b><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font></span>)</span>;<br/><a NAME="E3:15_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E22">A11</font></i>: 
<font color="Maroon" title="c7">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="topreal1.html#E5:15_1_1_1_1_1_1"><i><font color="Green" title="E19">A8</font></i></a>, <a class="txt" href="topreal1.html#E2:15_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E21">A10</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E4:15_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c7">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="topreal1.html#E1:15_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E20">A9</font></i></a>, <a class="txt" href="topreal1.html#E2:15_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E21">A10</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E5:15_1_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E23">A12</font></i>: 
<font color="Maroon" title="c7">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="topreal1.html#E1:15_1_1_1_1"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="xcmplx_1.html#T107" title="XCMPLX_1:th.107">XCMPLX_1:107</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p1</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">p2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="topreal1.html#E4:15_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="euclid.html#T36" title="EUCLID:th.36">EUCLID:36</a></i>
<br/>.= 
<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" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p1</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">p2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T30" title="EUCLID:th.30">EUCLID:30</a></i>
<br/>.= 
<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" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p1</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">p2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T34" title="EUCLID:th.34">EUCLID:34</a></i>
<br/>.= 
<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" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p1</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">p2</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T34" title="EUCLID:th.34">EUCLID:34</a></i>
<br/>.= 
<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" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">p2</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T37" title="EUCLID:th.37">EUCLID:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">p2</font></span>)</span>
<b>by </b><i><a class="txt" href="topreal1.html#E1:15_1_1_1_1"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="xcmplx_1.html#T110" title="XCMPLX_1:th.110">XCMPLX_1:110</a></i>
<br/>.= 
<font color="Maroon" title="c6">q</font>
<b>by </b><i><a class="txt" href="topreal1.html#E1:15_1_1_1_1"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="topreal1.html#E4:15_1_1_1_1_1_1"><i><font color="Green" title="E18">A7</font></i></a>, <a class="ref" href="xcmplx_1.html#T110" title="XCMPLX_1:th.110">XCMPLX_1:110</a></i>
;<br/><a NAME="E7:15_1_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">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" title="REAL_1:func.5">-</a> <font color="Olive" title="b1">lambda</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">lambda</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> where <font color="Olive" title="b1">lambda</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">lambda</font> &amp; <font color="Olive" title="b1">lambda</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="topreal1.html#E3:15_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E22">A11</font></i></a>, <a class="txt" href="topreal1.html#E5:15_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E23">A12</font></i></a></i>;<br/><a NAME="E8:15_1_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c2">p</font>
 ;<br/><b>hence </b><a NAME="E9:15_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:15_1_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E20">A13</font></i>: 
not <font color="Maroon" title="c7">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c8">bp</font> = 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">b</font>;<br/><b>set </b><font color="Maroon" title="c9">rp</font> = 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font>;<br/><a NAME="E2:15_1_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E21">A14</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">b</font>
 <b>by </b><i><a class="txt" href="topreal1.html#E5:15_1_1_1_1_1_1"><i><font color="Green" title="E19">A8</font></i></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/><a NAME="E3:15_1_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E22">A15</font></i>: 
1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="txt" href="topreal1.html#E1:15_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E20">A13</font></i></a>, <a class="ref" href="real_2.html#T105" title="REAL_2:th.105">REAL_2:105</a></i>;<br/><a NAME="E4:15_1_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E23">A16</font></i>: 
1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E5:15_1_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E24">A17</font></i>: 
( 0 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font> &amp; 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 <b>by </b><i><a class="txt" href="topreal1.html#E1:15_1_1_1_1"><i><font color="Green" title="E16">A5</font></i></a></i>;<br/><a NAME="E6:15_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c5">r</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><a NAME="E7:15_1_1_1_1_1_1_1_1_2"/><b>then </b>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="txt" href="topreal1.html#E6:15_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E8:15_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E25">A18</font></i>: 
1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="real_2.html#T125" title="REAL_2:th.125">REAL_2:125</a></i>;<br/><b>set </b><font color="Maroon" title="c10">x</font> = <span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span>;<br/><a NAME="E9:15_1_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E26">A19</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="topreal1.html#E2:15_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E21">A14</font></i></a>, <a class="txt" href="topreal1.html#E8:15_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E25">A18</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E10:15_1_1_1_1_1_1_1_1_2"/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="topreal1.html#E3:15_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E22">A15</font></i></a>, <a class="txt" href="topreal1.html#E8:15_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E25">A18</font></i></a>, <a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E11:15_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E27">A20</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="topreal1.html#E5:15_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E24">A17</font></i></a>, <a class="ref" href="xcmplx_1.html#T107" title="XCMPLX_1:th.107">XCMPLX_1:107</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">p2</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</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" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">p2</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</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" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <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" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p1</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="topreal1.html#E4:15_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="euclid.html#T36" title="EUCLID:th.36">EUCLID:36</a></i>
<br/>.= 
<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" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">p2</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <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" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">p2</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</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" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T30" title="EUCLID:th.30">EUCLID:30</a></i>
<br/>.= 
<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" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">p2</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</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">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</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" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T34" title="EUCLID:th.34">EUCLID:34</a></i>
<br/>.= 
<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" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">p2</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</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">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</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" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p1</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T34" title="EUCLID:th.34">EUCLID:34</a></i>
<br/>.= 
<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" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">p2</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</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" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p1</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T37" title="EUCLID:th.37">EUCLID:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">p2</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p1</font></span>)</span>
<b>by </b><i><a class="txt" href="topreal1.html#E1:15_1_1_1_1"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="topreal1.html#E4:15_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E23">A16</font></i></a>, <a class="ref" href="xcmplx_1.html#T110" title="XCMPLX_1:th.110">XCMPLX_1:110</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">b</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">p2</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p1</font></span>)</span>
<b>by </b><i><a class="txt" href="topreal1.html#E5:15_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E24">A17</font></i></a>, <a class="ref" href="xcmplx_1.html#T110" title="XCMPLX_1:th.110">XCMPLX_1:110</a></i>
<br/>.= 
<font color="Maroon" title="c6">q</font>
<b>by </b><i><a class="txt" href="topreal1.html#E4:15_1_1_1_1_1_1"><i><font color="Green" title="E18">A7</font></i></a></i>
;<br/><a NAME="E13:15_1_1_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">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" title="REAL_1:func.5">-</a> <font color="Olive" title="b1">lambda</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">p2</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">lambda</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> where <font color="Olive" title="b1">lambda</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">lambda</font> &amp; <font color="Olive" title="b1">lambda</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="txt" href="topreal1.html#E9:15_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E26">A19</font></i></a>, <a class="txt" href="topreal1.html#E11:15_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E27">A20</font></i></a></i>;<br/><a NAME="E14:15_1_1_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font>
 <b>by </b><i><a class="ref" href="topreal1.html#D3" target="_self" title="TOPREAL1:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E15:15_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:15_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font></span>)</span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:15_1_1_1_1_1"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:15_1_1_1_1"/>
( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> implies  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="topreal1.html#E3:15_1_1_1_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_2"><b>suppose </b></a><a NAME="E1:15_1_1_1_2"/><i><font color="Green" title="E16">A21</font></i>: 
(  not 0 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">r</font> or  not <font color="Maroon" title="c5">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1_1_1_2_1_1"/>
( <font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><b>by </b><i><a class="txt" href="topreal1.html#E1:15_1_1_1_2"><i><font color="Green" title="E16">A21</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:15_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>then </b><i><font color="Green" title="E17">A22</font></i>: <font color="Maroon" title="c2">p</font> = 
<span class="p1">(<span class="default">1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
<b>by </b><i><a class="txt" href="topreal1.html#E4:15_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>
<br/>.= 
<font color="Maroon" title="c3">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>
<br/>.= 
<font color="Maroon" title="c3">p1</font>
<b>by </b><i><a class="ref" href="euclid.html#T31" title="EUCLID:th.31">EUCLID:31</a></i>
;<br/><a NAME="E3:15_1_1_1_2_1_1_1"/><b>then </b>
(  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font> )
 <b>by </b><i><a class="ref" href="topreal1.html#T6" target="_self" title="TOPREAL1:th.6">Th6</a>, <a class="ref" href="topreal1.html#T7" target="_self" title="TOPREAL1:th.7">Th7</a></i>;<br/><a NAME="E4:15_1_1_1_2_1_1_1"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><b>hence </b><a NAME="E5:15_1_1_1_2_1_1_1"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font></span>)</span>
 <b>by </b><i><a class="txt" href="topreal1.html#E2:15_1_1_1_2_1_1_1"><i><font color="Green" title="E17">A22</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:15_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>then </b><i><font color="Green" title="E17">A23</font></i>: <font color="Maroon" title="c2">p</font> = 
<span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default">1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">p2</font></span>)</span>
<b>by </b><i><a class="txt" href="topreal1.html#E4:15_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c4">p2</font>
<b>by </b><i><a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>
<br/>.= 
<font color="Maroon" title="c4">p2</font>
<b>by </b><i><a class="ref" href="euclid.html#T31" title="EUCLID:th.31">EUCLID:31</a></i>
;<br/><a NAME="E3:15_1_1_1_2_1_1_2"/><b>then </b>
(  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c2">p</font> )
 <b>by </b><i><a class="ref" href="topreal1.html#T6" target="_self" title="TOPREAL1:th.6">Th6</a>, <a class="ref" href="topreal1.html#T7" target="_self" title="TOPREAL1:th.7">Th7</a></i>;<br/><a NAME="E4:15_1_1_1_2_1_1_2"/><b>then </b>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><b>hence </b><a NAME="E5:15_1_1_1_2_1_1_2"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font></span>)</span>
 <b>by </b><i><a class="txt" href="topreal1.html#E2:15_1_1_1_2_1_1_2"><i><font color="Green" title="E17">A23</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:15_1_1_1_2"/>
( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> implies  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font></span>)</span> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:15_1"/>
( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> implies  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font></span>)</span> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:15"/>
( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> implies  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">p2</font></span>)</span> )
 ;<br/>


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