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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c3">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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/><b>let </b><font color="Maroon" title="c4">w1</font>, <font color="Maroon" title="c5">w7</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/>







<b>assume </b><a NAME="E1:58"/><i><font color="Green" title="E50">A1</font></i>: 
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Maroon" title="c3">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <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> : <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">a</font> </span>}</span>  &amp; <font color="Maroon" title="c4">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> &amp; <font color="Maroon" title="c5">w7</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> &amp;  ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c4">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">w7</font> or <font color="Maroon" title="c5">w7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">w1</font> ) )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">r8</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:58"/><i><font color="Green" title="E51">A2</font></i>: 
( <font color="Maroon" title="c4">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r8</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">w7</font> or <font color="Maroon" title="c5">w7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r8</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">w1</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">y1</font> = <font color="Maroon" title="c4">w1</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="ref" href="euclid.html#T25" title="EUCLID:th.25">EUCLID:25</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:58_1"/>
( <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1"><b>suppose </b></a><a NAME="E1:58_1_1"/><i><font color="Green" title="E52">A3</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:58_1_1_1"/>
<font color="Maroon" title="c4">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font>
 ;<br/><a NAME="E2:58_1_1_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c4">w1</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="toprns_1.html#T24" title="TOPRNS_1:th.24">TOPRNS_1:24</a></i>;<br/><a NAME="E3:58_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <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> : <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">a</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:58_1_1"><i><font color="Green" title="E52">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E4:58_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:58"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c8">y</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E4:58_1_1"/><i><font color="Green" title="E53">A4</font></i>: 
for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> ( not <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">y1</font> &amp; not <font color="Maroon" title="c7">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c8">y</font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:58"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="jordan2c.html#T55" target="_self" title="JORDAN2C:th.55">Th55</a></i>;<br/><b>set </b><font color="Maroon" title="c9">y4</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c8">y</font>;<br/><b>reconsider </b><font color="Maroon" title="c10">w4</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c8">y</font> as   <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> <b>by </b><i><a class="ref" href="euclid.html#T25" title="EUCLID:th.25">EUCLID:25</a></i>;<br/><a NAME="E6:58_1_1"/><i><font color="Green" title="E54">A5</font></i>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div><i><font color="Green" title="E55">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:58_1_1_2"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E2:58_1_1_2"/><b>then </b><i><font color="Green" title="E56">A7</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="euclid.html#T11" title="EUCLID:th.11">EUCLID:11</a></i>;<br/><a NAME="E3:58_1_1_2"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>;<br/><b>hence </b><a NAME="E4:58_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:58_1_1"><i><font color="Green" title="E53">A4</font></i></a>, <a class="txt" href="jordan2c.html#E2:58_1_1_2"><i><font color="Green" title="E56">A7</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E8:58_1_1"/><b>then </b><i><font color="Green" title="E56">A8</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:58_1_1"><i><font color="Green" title="E52">A3</font></i></a>, <a class="txt" href="jordan2c.html#E6:58_1_1"><i><font color="Green" title="E54">A5</font></i></a>, <a class="ref" href="real_2.html#T127" title="REAL_2:th.127">REAL_2:127</a></i>;<br/><i><font color="Green" title="E57">A9</font></i>: <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c10">w4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T14" title="EUCLID:th.14">EUCLID:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="txt" href="jordan2c.html#E8:58_1_1"><i><font color="Green" title="E56">A8</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="txt" href="jordan2c.html#E7:58_1_1"><i><font color="Green" title="E55">A6</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>
;<br/><div><i><font color="Green" title="E58">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:58_1_1_4"/>
<font color="Maroon" title="c10">w4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <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> : <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">a</font> </span>}</span> 
 ;<br/><a NAME="E2:58_1_1_4"/><b>then </b>
 ex <font color="Olive" title="b1">q</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> st <br/>( <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">w4</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">a</font> )
 ;<br/><b>hence </b><a NAME="E3:58_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E9:58_1_1"><i><font color="Green" title="E57">A9</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E11:58_1_1"/><b>then </b><i><font color="Green" title="E59">A11</font></i>: 
<font color="Maroon" title="c10">w4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:58"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><div><i><font color="Green" title="E60">A12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_5"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c11">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><a NAME="E2:58_1_1_5"/><i><font color="Green" title="E61">A13</font></i>: 
( <font color="Maroon" title="c4">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">w4</font> or <font color="Maroon" title="c10">w4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">w1</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c12">y'</font> = <font color="Maroon" title="c8">y</font>, <font color="Maroon" title="c13">y1'</font> = <font color="Maroon" title="c7">y1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/><a NAME="E4:58_1_1_5"/>
( <font color="Maroon" title="c7">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c8">y</font> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c12">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">y1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E2:58_1_1_5"><i><font color="Green" title="E61">A13</font></i></a>, <a class="ref" href="rvsum_1.html#T71" title="RVSUM_1:th.71">RVSUM_1:71</a></i>;<br/><a NAME="E5:58_1_1_5"/><b>then </b>
( <font color="Maroon" title="c7">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c8">y</font> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c11">r</font></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c13">y1'</font> )
 <b>by </b><i><a class="ref" href="rvsum_1.html#T71" title="RVSUM_1:th.71">RVSUM_1:71</a></i>;<br/><a NAME="E6:58_1_1_5"/><b>then </b><i><font color="Green" title="E62">A14</font></i>: 
( <font color="Maroon" title="c7">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c12">y'</font> or 1 <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c11">r</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">y1</font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E8:58_1_1"><i><font color="Green" title="E56">A8</font></i></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:58_1_1_5_1"/>
( <font color="Maroon" title="c7">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c8">y</font> or <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c11">r</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">y1</font> )
 </span><b>by </b><i><a class="txt" href="jordan2c.html#E6:58_1_1_5"><i><font color="Green" title="E62">A14</font></i></a>, <a class="ref" href="rvsum_1.html#T74" title="RVSUM_1:th.74">RVSUM_1:74</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_5_1_1"><b>suppose </b></a><a NAME="E1:58_1_1_5_1_1"/>
<font color="Maroon" title="c7">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">r</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c8">y</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:58_1_1_5_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:58_1_1"><i><font color="Green" title="E53">A4</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_5_1_2"><b>suppose </b></a><a NAME="E1:58_1_1_5_1_2"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c8">y</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c11">r</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">y1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:58_1_1_5_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:58_1_1"><i><font color="Green" title="E53">A4</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c11">w2</font>, <font color="Maroon" title="c12">w3</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E14:58_1_1"/><i><font color="Green" title="E61">A15</font></i>: 
( <font color="Maroon" title="c11">w2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> &amp; <font color="Maroon" title="c12">w3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c4">w1</font>,<font color="Maroon" title="c11">w2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Q</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c11">w2</font>,<font color="Maroon" title="c12">w3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Q</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c12">w3</font>,<font color="Maroon" title="c10">w4</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Q</font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:58"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="jordan2c.html#E11:58_1_1"><i><font color="Green" title="E59">A11</font></i></a>, <a class="ref" href="jordan2c.html#T50" target="_self" title="JORDAN2C:th.50">Th50</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_6"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c13">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><a NAME="E2:58_1_1_6"/><i><font color="Green" title="E62">A16</font></i>: 
( <font color="Maroon" title="c10">w4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">r1</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">w7</font> or <font color="Maroon" title="c5">w7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">r1</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">w4</font> )
 ;<br/><div><i><font color="Green" title="E63">A17</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:58_1_1_6_1"/>
<font color="Maroon" title="c13">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E2:58_1_1_6_1"/><b>then </b>
( <font color="Maroon" title="c10">w4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font> or <font color="Maroon" title="c5">w7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E2:58_1_1_6"><i><font color="Green" title="E62">A16</font></i></a>, <a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>;<br/><a NAME="E3:58_1_1_6_1"/><b>then </b>
( <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c10">w4</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">w7</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="toprns_1.html#T24" title="TOPRNS_1:th.24">TOPRNS_1:24</a></i>;<br/><a NAME="E4:58_1_1_6_1"/><b>then </b>
( <font color="Maroon" title="c10">w4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <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> : <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">a</font> </span>}</span>  or <font color="Maroon" title="c5">w7</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q2</font> where <font color="Olive" title="b1">q2</font> is   <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> : <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q2</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">a</font> </span>}</span>  )
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:58_1_1"><i><font color="Green" title="E52">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E5:58_1_1_6_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:58"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="jordan2c.html#E10:58_1_1"><i><font color="Green" title="E58">A10</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_6_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_6_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:58_1_1_6_2_1"/>
( <font color="Maroon" title="c4">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r8</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">w7</font> or <font color="Maroon" title="c5">w7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r8</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">w1</font> )
 </span><b>by </b><i><a class="txt" href="jordan2c.html#E3:58"><i><font color="Green" title="E51">A2</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_6_2_1_1"><b>case </b></a><a NAME="E1:58_1_1_6_2_1_1"/><i><font color="Green" title="E64">A18</font></i>: 
<font color="Maroon" title="c4">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r8</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">w7</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_6_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_6_2_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="E1:58_1_1_6_2_1_1_1_1"/>
( <font color="Maroon" title="c10">w4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">r1</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">w7</font> or <font color="Maroon" title="c5">w7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">r1</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">w4</font> )
 </span><b>by </b><i><a class="txt" href="jordan2c.html#E2:58_1_1_6"><i><font color="Green" title="E62">A16</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_6_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:58_1_1_6_2_1_1_1_1_1"/>
<font color="Maroon" title="c10">w4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">r1</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">w7</font>
 ;<br/><div class="add"><a NAME="E2:58_1_1_6_2_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">r1</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">w4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">r1</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c13">r1</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">w7</font>
 <b>by </b><i><a class="ref" href="euclid.html#T34" title="EUCLID:th.34">EUCLID:34</a></i>;<br/><a NAME="E3:58_1_1_6_2_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">r1</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">w4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">w7</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:58_1_1_6"><i><font color="Green" title="E63">A17</font></i></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></i>;<br/><a NAME="E4:58_1_1_6_2_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">r1</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">w4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">w7</font>
 <b>by </b><i><a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>;<br/><a NAME="E5:58_1_1_6_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">r8</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">r1</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">w4</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:58_1_1_6_2_1_1"><i><font color="Green" title="E64">A18</font></i></a>, <a class="ref" href="euclid.html#T34" title="EUCLID:th.34">EUCLID:34</a></i>;<br/><b>hence </b><a NAME="E6:58_1_1_6_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E12:58_1_1"><i><font color="Green" title="E60">A12</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_6_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:58_1_1_6_2_1_1_1_1_2"/>
<font color="Maroon" title="c5">w7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">r1</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">w4</font>
 ;<br/><div class="add"><a NAME="E2:58_1_1_6_2_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">r1</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">w7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">r1</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c13">r1</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">w4</font>
 <b>by </b><i><a class="ref" href="euclid.html#T34" title="EUCLID:th.34">EUCLID:34</a></i>;<br/><a NAME="E3:58_1_1_6_2_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">r1</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">w7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">w4</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:58_1_1_6"><i><font color="Green" title="E63">A17</font></i></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></i>;<br/><a NAME="E4:58_1_1_6_2_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">r1</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">w7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">w4</font>
 <b>by </b><i><a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>;<br/><a NAME="E5:58_1_1_6_2_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">r1</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">w4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">r1</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">r1</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">w7</font>
 <b>by </b><i><a class="ref" href="euclid.html#T34" title="EUCLID:th.34">EUCLID:34</a></i>;<br/><a NAME="E6:58_1_1_6_2_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">r1</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">w4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">w7</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:58_1_1_6"><i><font color="Green" title="E63">A17</font></i></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></i>;<br/><a NAME="E7:58_1_1_6_2_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">r1</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">w4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">w7</font>
 <b>by </b><i><a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>;<br/><a NAME="E8:58_1_1_6_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">r8</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">r1</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">w4</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:58_1_1_6_2_1_1"><i><font color="Green" title="E64">A18</font></i></a>, <a class="ref" href="euclid.html#T34" title="EUCLID:th.34">EUCLID:34</a></i>;<br/><b>hence </b><a NAME="E9:58_1_1_6_2_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E12:58_1_1"><i><font color="Green" title="E60">A12</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:58_1_1_6_2_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_6_2_1_2"><b>case </b></a><a NAME="E1:58_1_1_6_2_1_2"/><i><font color="Green" title="E64">A19</font></i>: 
<font color="Maroon" title="c5">w7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r8</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">w1</font>
 ;<br/><div class="add"><a NAME="E2:58_1_1_6_2_1_2"/><b>then </b><i><font color="Green" title="E65">A20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">r8</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">w7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">r8</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c6">r8</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">w1</font>
 <b>by </b><i><a class="ref" href="euclid.html#T34" title="EUCLID:th.34">EUCLID:34</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_6_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:58_1_1_6_2_1_2_1"/>
<font color="Maroon" title="c6">r8</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E2:58_1_1_6_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c5">w7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:58_1_1_6_2_1_2"><i><font color="Green" title="E64">A19</font></i></a>, <a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>;<br/><a NAME="E3:58_1_1_6_2_1_2_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">w7</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="toprns_1.html#T24" title="TOPRNS_1:th.24">TOPRNS_1:24</a></i>;<br/><a NAME="E4:58_1_1_6_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c5">w7</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <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> : <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">a</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:58_1_1"><i><font color="Green" title="E52">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E5:58_1_1_6_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:58"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><a NAME="E4:58_1_1_6_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">r8</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">w7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">w1</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E2:58_1_1_6_2_1_2"><i><font color="Green" title="E65">A20</font></i></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></i>;<br/><a NAME="E5:58_1_1_6_2_1_2"/><b>then </b><i><font color="Green" title="E66">A21</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">r8</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">w7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w1</font>
 <b>by </b><i><a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_6_2_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_6_2_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:58_1_1_6_2_1_2_2_1"/>
( <font color="Maroon" title="c10">w4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">r1</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">w7</font> or <font color="Maroon" title="c5">w7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">r1</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">w4</font> )
 </span><b>by </b><i><a class="txt" href="jordan2c.html#E2:58_1_1_6"><i><font color="Green" title="E62">A16</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_6_2_1_2_2_1_1"><b>case </b></a><a NAME="E1:58_1_1_6_2_1_2_2_1_1"/>
<font color="Maroon" title="c10">w4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">r1</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">w7</font>
 ;<br/><div class="add"><a NAME="E2:58_1_1_6_2_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">r1</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">w4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">r1</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c13">r1</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">w7</font>
 <b>by </b><i><a class="ref" href="euclid.html#T34" title="EUCLID:th.34">EUCLID:34</a></i>;<br/><a NAME="E3:58_1_1_6_2_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">r1</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">w4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">w7</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:58_1_1_6"><i><font color="Green" title="E63">A17</font></i></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></i>;<br/><a NAME="E4:58_1_1_6_2_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">r1</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">w4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">w7</font>
 <b>by </b><i><a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>;<br/><a NAME="E5:58_1_1_6_2_1_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c4">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">r8</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">r1</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">w4</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E5:58_1_1_6_2_1_2"><i><font color="Green" title="E66">A21</font></i></a>, <a class="ref" href="euclid.html#T34" title="EUCLID:th.34">EUCLID:34</a></i>;<br/><b>hence </b><a NAME="E6:58_1_1_6_2_1_2_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E12:58_1_1"><i><font color="Green" title="E60">A12</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_6_2_1_2_2_1_2"><b>case </b></a><a NAME="E1:58_1_1_6_2_1_2_2_1_2"/>
<font color="Maroon" title="c5">w7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">r1</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">w4</font>
 ;<br/><div class="add"><a NAME="E2:58_1_1_6_2_1_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">r1</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">w7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">r1</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c13">r1</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">w4</font>
 <b>by </b><i><a class="ref" href="euclid.html#T34" title="EUCLID:th.34">EUCLID:34</a></i>;<br/><a NAME="E3:58_1_1_6_2_1_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">r1</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">w7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">w4</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:58_1_1_6"><i><font color="Green" title="E63">A17</font></i></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></i>;<br/><a NAME="E4:58_1_1_6_2_1_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">r1</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">w7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">w4</font>
 <b>by </b><i><a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>;<br/><a NAME="E5:58_1_1_6_2_1_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">r1</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">w4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">r1</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">r1</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">w7</font>
 <b>by </b><i><a class="ref" href="euclid.html#T34" title="EUCLID:th.34">EUCLID:34</a></i>;<br/><a NAME="E6:58_1_1_6_2_1_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">r1</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">w4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">w7</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:58_1_1_6"><i><font color="Green" title="E63">A17</font></i></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></i>;<br/><a NAME="E7:58_1_1_6_2_1_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">r1</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">w4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">w7</font>
 <b>by </b><i><a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>;<br/><a NAME="E8:58_1_1_6_2_1_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c4">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">r8</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">r1</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> </span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">w4</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E5:58_1_1_6_2_1_2"><i><font color="Green" title="E66">A21</font></i></a>, <a class="ref" href="euclid.html#T34" title="EUCLID:th.34">EUCLID:34</a></i>;<br/><b>hence </b><a NAME="E9:58_1_1_6_2_1_2_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E12:58_1_1"><i><font color="Green" title="E60">A12</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:58_1_1_6_2_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:58_1_1_6"/>
contradiction
 ;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c13">w2'</font>, <font color="Maroon" title="c14">w3'</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E17:58_1_1"/><i><font color="Green" title="E62">A22</font></i>: 
( <font color="Maroon" title="c13">w2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> &amp; <font color="Maroon" title="c14">w3'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c10">w4</font>,<font color="Maroon" title="c13">w2'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Q</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c13">w2'</font>,<font color="Maroon" title="c14">w3'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Q</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c14">w3'</font>,<font color="Maroon" title="c5">w7</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Q</font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:58"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="jordan2c.html#E11:58_1_1"><i><font color="Green" title="E59">A11</font></i></a>, <a class="ref" href="jordan2c.html#T50" target="_self" title="JORDAN2C:th.50">Th50</a></i>;<br/><b>thus </b><a NAME="E18:58_1_1"/>
 ex <font color="Olive" title="b1">w2</font>, <font color="Olive" title="b2">w3</font>, <font color="Olive" title="b3">w4</font>, <font color="Olive" title="b4">w5</font>, <font color="Olive" title="b5">w6</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> st <br/>( <font color="Olive" title="b1">w2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> &amp; <font color="Olive" title="b2">w3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> &amp; <font color="Olive" title="b3">w4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> &amp; <font color="Olive" title="b4">w5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> &amp; <font color="Olive" title="b5">w6</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c4">w1</font>,<font color="Olive" title="b1">w2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Q</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Olive" title="b1">w2</font>,<font color="Olive" title="b2">w3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Q</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Olive" title="b2">w3</font>,<font color="Olive" title="b3">w4</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Q</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Olive" title="b3">w4</font>,<font color="Olive" title="b4">w5</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Q</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Olive" title="b4">w5</font>,<font color="Olive" title="b5">w6</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Q</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Olive" title="b5">w6</font>,<font color="Maroon" title="c5">w7</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Q</font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E11:58_1_1"><i><font color="Green" title="E59">A11</font></i></a>, <a class="txt" href="jordan2c.html#E14:58_1_1"><i><font color="Green" title="E61">A15</font></i></a>, <a class="txt" href="jordan2c.html#E17:58_1_1"><i><font color="Green" title="E62">A22</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_2"><b>suppose </b></a><a NAME="E1:58_1_2"/><i><font color="Green" title="E52">A23</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:58_1_2"/><i><font color="Green" title="E53">A24</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="euclid.html#T25" title="EUCLID:th.25">EUCLID:25</a></i>;<br/><a NAME="E3:58_1_2"/>
 <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:58_1_2_1"/><i><font color="Green" title="E54">A25</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:58_1_2_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <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> : <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">a</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c9">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E3:58_1_2_1_1"/><i><font color="Green" title="E55">A26</font></i>: 
( <font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c9">q</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">a</font> )
 ;<br/><b>thus </b><a NAME="E4:58_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:58_1_2"><i><font color="Green" title="E52">A23</font></i></a>, <a class="txt" href="jordan2c.html#E3:58_1_2_1_1"><i><font color="Green" title="E55">A26</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:58_1_2_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:58"><i><font color="Green" title="E50">A1</font></i></a>, <a class="txt" href="jordan2c.html#E1:58_1_2_1"><i><font color="Green" title="E54">A25</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:58_1_2"/><b>then </b><i><font color="Green" title="E54">A27</font></i>: 
<font color="Maroon" title="c3">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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>
 <b>by </b><i><a class="txt" href="jordan2c.html#E2:58_1_2"><i><font color="Green" title="E53">A24</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><b>set </b><font color="Maroon" title="c8">w2</font> =  <a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font>;<br/><a NAME="E5:58_1_2"/><i><font color="Green" title="E55">A28</font></i>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c4">w1</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="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Q</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:58_1_2"><i><font color="Green" title="E54">A27</font></i></a></i>;<br/><a NAME="E6:58_1_2"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</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>,<font color="Maroon" title="c5">w7</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Q</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:58_1_2"><i><font color="Green" title="E54">A27</font></i></a></i>;<br/><b>hence </b><a NAME="E7:58_1_2"/>
 ex <font color="Olive" title="b1">w2</font>, <font color="Olive" title="b2">w3</font>, <font color="Olive" title="b3">w4</font>, <font color="Olive" title="b4">w5</font>, <font color="Olive" title="b5">w6</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> st <br/>( <font color="Olive" title="b1">w2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> &amp; <font color="Olive" title="b2">w3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> &amp; <font color="Olive" title="b3">w4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> &amp; <font color="Olive" title="b4">w5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> &amp; <font color="Olive" title="b5">w6</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c4">w1</font>,<font color="Olive" title="b1">w2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Q</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Olive" title="b1">w2</font>,<font color="Olive" title="b2">w3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Q</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Olive" title="b2">w3</font>,<font color="Olive" title="b3">w4</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Q</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Olive" title="b3">w4</font>,<font color="Olive" title="b4">w5</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Q</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Olive" title="b4">w5</font>,<font color="Olive" title="b5">w6</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Q</font> &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Olive" title="b5">w6</font>,<font color="Maroon" title="c5">w7</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Q</font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:58_1_2"><i><font color="Green" title="E54">A27</font></i></a>, <a class="txt" href="jordan2c.html#E5:58_1_2"><i><font color="Green" title="E55">A28</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
