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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">y</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>let </b><font color="Maroon">x</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>let </b><font color="Maroon">z</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><b>that </b><br/><a NAME="E1:52"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K3" target="_self">Sphere</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E2:52"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K3" target="_self">Sphere</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> or <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1"><b>suppose </b></a><a NAME="E1:52_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><a NAME="E2:52_1_1"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="topreal9.html#K4" target="_self">halfline</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i/>;<br/><a NAME="E3:52_1_1"/><i><font color="Green">E68</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="enumset1.html#T69">ENUMSET1:69</a></i>;<br/><b>hence </b><a NAME="E4:52_1_1"/>
for <font color="Olive">m</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">m</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal9.html#K4" target="_self">halfline</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" target="_self">Sphere</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span> holds <br/><font color="Olive">m</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/><b>let </b><font color="Maroon">m</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="E5:52_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K2">}</a></span>
 ;<br/><a NAME="E6:52_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:52_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal9.html#K4" target="_self">halfline</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" target="_self">Sphere</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2"><b>suppose </b></a><a NAME="E1:52_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:52_1_2_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal9.html#K4" target="_self">halfline</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" target="_self">Sphere</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span>
 ;<br/><a NAME="E2:52_1_2_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K3" target="_self">Sphere</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">w</font> = <font color="Maroon">m</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/>;<br/><a NAME="E4:52_1_2_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K4" target="_self">halfline</a> <font color="Maroon">y</font>,<font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">R</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E6:52_1_2_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">R</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">y</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="euclid.html#K18">*</a> <font color="Maroon">z</font></span>)</span>
 <b>and </b><br/><a NAME="E7:52_1_2_1"/><i><font color="Green">E74</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">R</font> = <font color="Maroon">R</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E9:52_1_2_1"/><i><font color="Green">E75</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">y</font>,<font color="Maroon">z</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"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">l</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">y</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Olive">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">z</font></span>)</span></span>)</span> where <font color="Olive">l</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="topreal1.html#D3">TOPREAL1:def 3</a></i>;<br/><a NAME="E10:52_1_2_1"/><i><font color="Green">E83</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">y</font>,<font color="Maroon">w</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"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">l</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">y</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Olive">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w</font></span>)</span></span>)</span> where <font color="Olive">l</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="topreal1.html#D3">TOPREAL1:def 3</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_2_1_1"/>
( <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> 1 or ( <font color="Maroon">R</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">R</font> <a href="xxreal_0.html#R1">&lt;</a> 1 ) or <font color="Maroon">R</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:52_1_2_1_1_1"/>
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:52_1_2_1_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:52_1_2_1_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:52_1_2_1_1_2"/>
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:52_1_2_1_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:52_1_2_1_1_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:52_1_2_1_1_3"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">R</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">R</font> <a href="xxreal_0.html#R1">&lt;</a> 1 )
 ;<br/><div class="add"><a NAME="E2:52_1_2_1_1_3"/><i><font color="Green">E85</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K1" target="_self">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E3:52_1_2_1_1_3"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">y</font>,<font color="Maroon">z</font>
 <b>by </b><i>, , <a class="ref" href="topreal9.html#T1" target="_self">Th1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_2_1_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_1_2_1_1_3_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K2">}</a></span>
 ;<br/><a NAME="E2:52_1_2_1_1_3_1"/><b>then </b>
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> or <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E3:52_1_2_1_1_3_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="topreal9.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div><a NAME="E5:52_1_2_1_1_3"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="topreal9.html#E1:52"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E6:52_1_2_1_1_3"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">w</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T2" target="_self">Th2</a>, </i>;<br/><b>hence </b><a NAME="E7:52_1_2_1_1_3"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1_4"><b>suppose </b></a><a NAME="E1:52_1_2_1_1_4"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">R</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:52_1_2_1_1_4"/><b>then </b><i><font color="Green">E120</font></i>: 
1 <a href="real_1.html#K6">/</a> <font color="Maroon">R</font> <a href="xxreal_0.html#R1">&lt;</a> 1
 <b>by </b><i><a class="ref" href="square_1.html#T2">SQUARE_1:2</a></i>;<br/><a NAME="E3:52_1_2_1_1_4"/><i><font color="Green">E121</font></i>: 
1 <a href="real_1.html#K6">/</a> <font color="Maroon">R</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="topreal9.html#T1" target="_self">Th1</a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><i><font color="Green">E122</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">y</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">R</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">y</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">R</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">R</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">R</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="euclid.html#K18">*</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="euclid.html#T36">EUCLID:36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">y</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">R</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">R</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">R</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">R</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">z</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T34">EUCLID:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">y</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">R</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">R</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default">1 <a href="euclid.html#K18">*</a> <font color="Maroon">z</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="topreal9.html#T1" target="_self">Th1</a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">y</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">R</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">R</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">y</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">R</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">R</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="euclid.html#T30">EUCLID:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">y</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">R</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="euclid.html#T34">EUCLID:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">R</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">R</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">y</font></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="euclid.html#T37">EUCLID:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">R</font></span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">R</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">y</font></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">z</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">R</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">R</font></span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">y</font></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="topreal9.html#T1" target="_self">Th1</a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
<font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>
;<br/><a NAME="E5:52_1_2_1_1_4"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">y</font>,<font color="Maroon">w</font>
 <b>by </b><i>, <a class="ref" href="topreal9.html#T2" target="_self">Th2</a>, <a class="txt" href="topreal9.html#E1:52"><i><font color="Green">E42</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_2_1_1_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_1_2_1_1_4_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">w</font></span><a href="tarski.html#K2">}</a></span>
 ;<br/><a NAME="E2:52_1_2_1_1_4_2"/><b>then </b>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E3:52_1_2_1_1_4_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="topreal9.html#T2" target="_self">Th2</a>, <a class="ref" href="topreal9.html#T3" target="_self">Th3</a>, </i>;<br/></div><b>end;</b></div><a NAME="E7:52_1_2_1_1_4"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">y</font>,<font color="Maroon">w</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">w</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="topreal9.html#E2:52"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E8:52_1_2_1_1_4"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">y</font>,<font color="Maroon">w</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">w</font></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K1" target="_self">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E9:52_1_2_1_1_4"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T4" target="_self">Th4</a>, </i>;<br/><b>hence </b><a NAME="E10:52_1_2_1_1_4"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div><b>let </b><font color="Maroon">m</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="E3:52_1_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K2">}</a></span>
 ;<br/><a NAME="E4:52_1_2"/><b>then </b><i><font color="Green">E125</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> or <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E5:52_1_2"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K4" target="_self">halfline</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K4" target="_self">halfline</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E6:52_1_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal9.html#K4" target="_self">halfline</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3" target="_self">Sphere</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
