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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">Q</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">w1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">w7</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:61"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &amp; <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> </span>}</span>  &amp; <font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">w7</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp;  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w7</font> or <font color="Maroon">w7</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w1</font> ) )
 ;<br/>

<b>then consider </b><font color="Maroon">r8</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:61"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r8</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w7</font> or <font color="Maroon">w7</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r8</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w1</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">w1</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:61_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1"><b>suppose </b></a><a NAME="E1:61_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_1_1_1"/>
<font color="Maroon">w1</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E2:61_1_1_1"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">w1</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="toprns_1.html#T24">TOPRNS_1:24</a></i>;<br/><a NAME="E3:61_1_1_1"/><b>then </b>
<font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> </span>}</span> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:61_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">y</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E4:61_1_1"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> holds <br/> ( not <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="euclid.html#K9">*</a> <font color="Maroon">y1</font> &amp; not <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="euclid.html#K9">*</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="jordan2c.html#T4" target="_self">Th4</a></i>;<br/><b>set </b><font color="Maroon">y4</font> = <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K12">/</a> <span class="p2"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">y</font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">y</font>;<br/><b>reconsider </b><font color="Maroon">w4</font> = <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K12">/</a> <span class="p2"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">y</font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">y</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/><a NAME="E6:61_1_1"/><i><font color="Green">E59</font></i>: 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">y</font></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="euclid.html#T12">EUCLID:12</a></i>;<br/><div><i><font color="Green">E60</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_1_1_2"/>
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">y</font></span><a href="euclid.html#K12">.|</a></span> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:61_1_1_2"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="euclid.html#T11">EUCLID:11</a></i>;<br/>0 <a href="euclid.html#K9">*</a> <font color="Maroon">y1</font> = 
0 <a href="euclid.html#K18">*</a> <font color="Maroon">w1</font>
<b>by </b><i><a class="ref" href="euclid.html#D11">EUCLID:def 11</a></i>
<br/>.= 
 <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
 <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font>

;<br/><b>hence </b><a NAME="E4:61_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:61"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan2c.html#E6:61_1_1"><i><font color="Green">E59</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E8:61_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">a</font> <a href="binop_2.html#K12">/</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">y</font></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="txt" href="jordan2c.html#E1:61_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><i><font color="Green">E64</font></i>: <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">w4</font></span><a href="toprns_1.html#K5">.|</a></span> = 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K12">/</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">y</font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">y</font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="jgraph_1.html#D5">JGRAPH_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K12">/</a> <span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">y</font></span><a href="euclid.html#K12">.|</a></span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">y</font></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T14">EUCLID:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K12">/</a> <span class="p2"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">y</font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">y</font></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="txt" href="jordan2c.html#E6:61_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="txt" href="jordan2c.html#E4:61_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
;<br/><div><i><font color="Green">E65</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_1_1_4"/>
<font color="Maroon">w4</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> </span>}</span> 
 ;<br/><a NAME="E2:61_1_1_4"/><b>then </b>
 ex <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> st <br/>( <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w4</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> )
 ;<br/><b>hence </b><a NAME="E3:61_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E7:61_1_1"><i><font color="Green">E60</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E11:61_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">w4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><div><i><font color="Green">E67</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_5"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><a NAME="E1:61_1_1_5"/><i><font color="Green">E130</font></i>: 
( <font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w4</font> or <font color="Maroon">w4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w1</font> )
 ;<br/><b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font>, <font color="Maroon">y1'</font> = <font color="Maroon">y1</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  ;<br/><a NAME="E3:61_1_1_5"/>
( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K12">/</a> <span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">y</font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">y</font></span>)</span> or <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K12">/</a> <span class="p2"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">y</font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="euclid.html#K9">*</a> <font color="Maroon">y1</font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E9:61_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="euclid.html#D11">EUCLID:def 11</a></i>;<br/><a NAME="E4:61_1_1_5"/><b>then </b>
( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K12">/</a> <span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">y</font></span><a href="euclid.html#K12">.|</a></span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">y</font> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K12">/</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">y</font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K12">/</a> <span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">y</font></span><a href="euclid.html#K12">.|</a></span></span>)</span></span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K12">/</a> <span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">y</font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K9">*</a> <font color="Maroon">y1</font></span>)</span> )
 <b>by </b><i><a class="ref" href="rvsum_1.html#T71">RVSUM_1:71</a></i>;<br/><a NAME="E5:61_1_1_5"/><b>then </b>
( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K12">/</a> <span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">y</font></span><a href="euclid.html#K12">.|</a></span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">y</font> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K12">/</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">y</font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K12">/</a> <span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">y</font></span><a href="euclid.html#K12">.|</a></span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K12">/</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">y</font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">y1'</font> )
 <b>by </b><i><a class="ref" href="rvsum_1.html#T71">RVSUM_1:71</a></i>;<br/><a NAME="E6:61_1_1_5"/><b>then </b>
( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K12">/</a> <span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">y</font></span><a href="euclid.html#K12">.|</a></span></span>)</span></span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">y'</font> or 1 <a href="euclid.html#K9">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K12">/</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">y</font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">y1</font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E6:61_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/><a NAME="E7:61_1_1_5"/><b>then </b><i><font color="Green">E131</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K12">/</a> <span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">y</font></span><a href="euclid.html#K12">.|</a></span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">y</font> or <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K12">/</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">y</font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">y1</font> )
 <b>by </b><i><a class="ref" href="rvsum_1.html#T74">RVSUM_1:74</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_1_5_1"/>
( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K12">/</a> <span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">y</font></span><a href="euclid.html#K12">.|</a></span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">y</font> or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K12">/</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">y</font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">y1</font> )
 </span><b>by </b><i><a class="txt" href="jordan2c.html#E10:61_1_1"><i><font color="Green">E65</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_5_1_1"><b>suppose </b></a><a NAME="E1:61_1_1_5_1_1"/>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K12">/</a> <span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">y</font></span><a href="euclid.html#K12">.|</a></span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:61_1_1_5_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:61"><i><font color="Green">E56</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_5_1_2"><b>suppose </b></a><a NAME="E1:61_1_1_5_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K12">/</a> <span class="p4"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">y</font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">y1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:61_1_1_5_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:61"><i><font color="Green">E56</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">w2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">w3</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E14:61_1_1"/><i><font color="Green">E152</font></i>: 
( <font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">w3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">w1</font>,<font color="Maroon">w2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">w2</font>,<font color="Maroon">w3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">w3</font>,<font color="Maroon">w4</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> )
 <b>by </b><i>, , <a class="ref" href="jordan2c.html#T1" target="_self">Th1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_6"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><a NAME="E1:61_1_1_6"/><i><font color="Green">E154</font></i>: 
( <font color="Maroon">w4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w7</font> or <font color="Maroon">w7</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w4</font> )
 ;<br/><div><i><font color="Green">E157</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_1_1_6_1"/>
<font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:61_1_1_6_1"/><b>then </b>
( <font color="Maroon">w4</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font> or <font color="Maroon">w7</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E10:61_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="euclid.html#T33">EUCLID:33</a></i>;<br/><a NAME="E3:61_1_1_6_1"/><b>then </b>
( <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">w4</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 0 or <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">w7</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="toprns_1.html#T24">TOPRNS_1:24</a></i>;<br/><a NAME="E4:61_1_1_6_1"/><b>then </b>
( <font color="Maroon">w4</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> </span>}</span>  or <font color="Maroon">w7</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q2</font> where <font color="Olive">q2</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> </span>}</span>  )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:61_1_1_6_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_6_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_1_6_2_1"/>
( <font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r8</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w7</font> or <font color="Maroon">w7</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r8</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w1</font> )
 </span><b>by </b><i><a class="ref" href="jordan2c.html#T7" target="_self">Th7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_6_2_1_1"><b>case </b></a><a NAME="E1:61_1_1_6_2_1_1"/><i><font color="Green">E158</font></i>: 
<font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r8</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w7</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_6_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_1_6_2_1_1_1_1"/>
( <font color="Maroon">w4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w7</font> or <font color="Maroon">w7</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w4</font> )
 </span><b>by </b><i><a class="txt" href="jordan2c.html#E10:61_1_1"><i><font color="Green">E65</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_6_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:61_1_1_6_2_1_1_1_1_1"/>
<font color="Maroon">w4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w7</font>
 ;<br/><div class="add"><a NAME="E2:61_1_1_6_2_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w4</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w7</font>
 <b>by </b><i><a class="ref" href="euclid.html#T34">EUCLID:34</a></i>;<br/><a NAME="E3:61_1_1_6_2_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w4</font> <a href="hidden.html#R1">=</a> 1 <a href="euclid.html#K18">*</a> <font color="Maroon">w7</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E11:61_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/><a NAME="E4:61_1_1_6_2_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w7</font>
 <b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>;<br/><a NAME="E5:61_1_1_6_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r8</font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="binop_2.html#K8">"</a> </span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w4</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E12:61_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="euclid.html#T34">EUCLID:34</a></i>;<br/><b>hence </b><a NAME="E6:61_1_1_6_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E8:61_1_1"><i><font color="Green">E63</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_6_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:61_1_1_6_2_1_1_1_1_2"/>
<font color="Maroon">w7</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w4</font>
 ;<br/><div class="add"><a NAME="E2:61_1_1_6_2_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w7</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w4</font>
 <b>by </b><i><a class="ref" href="euclid.html#T34">EUCLID:34</a></i>;<br/><a NAME="E3:61_1_1_6_2_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w7</font> <a href="hidden.html#R1">=</a> 1 <a href="euclid.html#K18">*</a> <font color="Maroon">w4</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E11:61_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/><a NAME="E4:61_1_1_6_2_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w7</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w4</font>
 <b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>;<br/><a NAME="E5:61_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">r1</font> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w4</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="binop_2.html#K8">"</a> </span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w7</font>
 <b>by </b><i><a class="ref" href="euclid.html#T34">EUCLID:34</a></i>;<br/><a NAME="E6:61_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">r1</font> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w4</font> <a href="hidden.html#R1">=</a> 1 <a href="euclid.html#K18">*</a> <font color="Maroon">w7</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E11:61_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/><a NAME="E7:61_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">r1</font> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w7</font>
 <b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>;<br/><a NAME="E8:61_1_1_6_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r8</font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w4</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E12:61_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="euclid.html#T34">EUCLID:34</a></i>;<br/><b>hence </b><a NAME="E9:61_1_1_6_2_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E8:61_1_1"><i><font color="Green">E63</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="E3:61_1_1_6_2_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_6_2_1_2"><b>case </b></a><a NAME="E1:61_1_1_6_2_1_2"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">w7</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r8</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w1</font>
 ;<br/><div class="add"><a NAME="E2:61_1_1_6_2_1_2"/><b>then </b><i><font color="Green">E160</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">r8</font> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w7</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r8</font> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">r8</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w1</font>
 <b>by </b><i><a class="ref" href="euclid.html#T34">EUCLID:34</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_6_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_1_1_6_2_1_2_1"/>
<font color="Maroon">r8</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:61_1_1_6_2_1_2_1"/><b>then </b>
<font color="Maroon">w7</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E12:61_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="euclid.html#T33">EUCLID:33</a></i>;<br/><a NAME="E3:61_1_1_6_2_1_2_1"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">w7</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="toprns_1.html#T24">TOPRNS_1:24</a></i>;<br/><a NAME="E4:61_1_1_6_2_1_2_1"/><b>then </b>
<font color="Maroon">w7</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> </span>}</span> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:61_1_1_6_2_1_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><a NAME="E4:61_1_1_6_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r8</font> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w7</font> <a href="hidden.html#R1">=</a> 1 <a href="euclid.html#K18">*</a> <font color="Maroon">w1</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T8" target="_self">Th8</a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/><a NAME="E5:61_1_1_6_2_1_2"/><b>then </b><i><font color="Green">E161</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">r8</font> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w7</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font>
 <b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_6_2_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_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="E1:61_1_1_6_2_1_2_2_1"/>
( <font color="Maroon">w4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w7</font> or <font color="Maroon">w7</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w4</font> )
 </span><b>by </b><i><a class="txt" href="jordan2c.html#E10:61_1_1"><i><font color="Green">E65</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_6_2_1_2_2_1_1"><b>case </b></a><a NAME="E1:61_1_1_6_2_1_2_2_1_1"/>
<font color="Maroon">w4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w7</font>
 ;<br/><div class="add"><a NAME="E2:61_1_1_6_2_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w4</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w7</font>
 <b>by </b><i><a class="ref" href="euclid.html#T34">EUCLID:34</a></i>;<br/><a NAME="E3:61_1_1_6_2_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w4</font> <a href="hidden.html#R1">=</a> 1 <a href="euclid.html#K18">*</a> <font color="Maroon">w7</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E11:61_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/><a NAME="E4:61_1_1_6_2_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w7</font>
 <b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>;<br/><a NAME="E5:61_1_1_6_2_1_2_2_1_1"/><b>then </b>
<font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r8</font> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="binop_2.html#K8">"</a> </span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w4</font>
 <b>by </b><i>, <a class="ref" href="euclid.html#T34">EUCLID:34</a></i>;<br/><b>hence </b><a NAME="E6:61_1_1_6_2_1_2_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E8:61_1_1"><i><font color="Green">E63</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_6_2_1_2_2_1_2"><b>case </b></a><a NAME="E1:61_1_1_6_2_1_2_2_1_2"/>
<font color="Maroon">w7</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w4</font>
 ;<br/><div class="add"><a NAME="E2:61_1_1_6_2_1_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w7</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w4</font>
 <b>by </b><i><a class="ref" href="euclid.html#T34">EUCLID:34</a></i>;<br/><a NAME="E3:61_1_1_6_2_1_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w7</font> <a href="hidden.html#R1">=</a> 1 <a href="euclid.html#K18">*</a> <font color="Maroon">w4</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E11:61_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/><a NAME="E4:61_1_1_6_2_1_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w7</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w4</font>
 <b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>;<br/><a NAME="E5:61_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">r1</font> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w4</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="binop_2.html#K8">"</a> </span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w7</font>
 <b>by </b><i><a class="ref" href="euclid.html#T34">EUCLID:34</a></i>;<br/><a NAME="E6:61_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">r1</font> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w4</font> <a href="hidden.html#R1">=</a> 1 <a href="euclid.html#K18">*</a> <font color="Maroon">w7</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E11:61_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/><a NAME="E7:61_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">r1</font> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w7</font>
 <b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>;<br/><a NAME="E8:61_1_1_6_2_1_2_2_1_2"/><b>then </b>
<font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r8</font> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="binop_2.html#K8">"</a> </span>)</span> <a href="binop_2.html#K8">"</a> </span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w4</font>
 <b>by </b><i>, <a class="ref" href="euclid.html#T34">EUCLID:34</a></i>;<br/><b>hence </b><a NAME="E9:61_1_1_6_2_1_2_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan2c.html#E8:61_1_1"><i><font color="Green">E63</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="E7:61_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="E4:61_1_1_6"/>
contradiction
 ;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">w2'</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">w3'</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E17:61_1_1"/><i><font color="Green">E162</font></i>: 
( <font color="Maroon">w2'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">w3'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">w4</font>,<font color="Maroon">w2'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">w2'</font>,<font color="Maroon">w3'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">w3'</font>,<font color="Maroon">w7</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> )
 <b>by </b><i>, , <a class="ref" href="jordan2c.html#T1" target="_self">Th1</a></i>;<br/><b>thus </b><a NAME="E18:61_1_1"/>
 ex <font color="Olive">w2</font>, <font color="Olive">w3</font>, <font color="Olive">w4</font>, <font color="Olive">w5</font>, <font color="Olive">w6</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> st <br/>( <font color="Olive">w2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Olive">w3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Olive">w4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Olive">w5</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Olive">w6</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">w1</font>,<font color="Olive">w2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Olive">w2</font>,<font color="Olive">w3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Olive">w3</font>,<font color="Olive">w4</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Olive">w4</font>,<font color="Olive">w5</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Olive">w5</font>,<font color="Olive">w6</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Olive">w6</font>,<font color="Maroon">w7</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> )
 <b>by </b><i>, <a class="txt" href="jordan2c.html#E9:61_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan2c.html#E10:61_1_1"><i><font color="Green">E65</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2"><b>suppose </b></a><a NAME="E1:61_1_2"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:61_1_2"/><i><font color="Green">E164</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/><a NAME="E3:61_1_2"/>
 <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:61_1_2_1"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">q</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E3:61_1_2_1_1"/><i><font color="Green">E166</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> )
 ;<br/><b>thus </b><a NAME="E4:61_1_2_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="toprns_1.html#T26">TOPRNS_1:26</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:61_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:61_1_2"/><b>then </b><i><font color="Green">E167</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:61"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>set </b><font color="Maroon">w2</font> =  <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font>;<br/><a NAME="E5:61_1_2"/><i><font color="Green">E168</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">w1</font>,<span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font>
 <b>by </b><i/>;<br/><a NAME="E6:61_1_2"/>
 <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">w7</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:61_1_2"/>
 ex <font color="Olive">w2</font>, <font color="Olive">w3</font>, <font color="Olive">w4</font>, <font color="Olive">w5</font>, <font color="Olive">w6</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> st <br/>( <font color="Olive">w2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Olive">w3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Olive">w4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Olive">w5</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Olive">w6</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">w1</font>,<font color="Olive">w2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Olive">w2</font>,<font color="Olive">w3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Olive">w3</font>,<font color="Olive">w4</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Olive">w4</font>,<font color="Olive">w5</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Olive">w5</font>,<font color="Olive">w6</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Olive">w6</font>,<font color="Maroon">w7</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
