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<div class="add">

<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">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">r</font> be  non <a href="xboole_0.html#V1">zero</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>





<a NAME="E1:38"/>
 <a href="topreal9.html#K1">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="topreal9.html#T16">TOPREAL9:16</a></i>;<br/>
<b>hence </b><a NAME="E2:38"/>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T31">TOPS_1:31</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_1"/>
(  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> is <a href="xboole_0.html#V1">empty</a> or  not  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1"><b>suppose </b></a><a NAME="E1:38_1_1"/><i><font color="Green">E69</font></i>: 
 <a href="topreal9.html#K1">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:38_1_1"/><b>then </b>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i><a class="ref" href="topreal9.html#T21">TOPREAL9:21</a></i>;<br/><a NAME="E3:38_1_1"/><b>then </b><i><font color="Green">E70</font></i>: 
 <a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="topreal9.html#T22">TOPREAL9:22</a></i>;<br/><a NAME="E4:38_1_1"/>
for <font color="Olive">A</font> being <a href="xboole_0.html#V1">empty</a> <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> holds   <a href="pre_topc.html#K4">Cl</a> <font color="Olive">A</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><b>hence </b><a NAME="E5:38_1_1"/>
 <a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2"><b>suppose </b></a><a NAME="E1:38_1_2"/><i><font color="Green">E74</font></i>: 
not  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>let </b><font color="Maroon">a</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="E2:38_1_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 ;<br/><b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</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> ;<br/><b>reconsider </b><font color="Maroon">ae</font> = <font color="Maroon">a</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/><a NAME="E5:38_1_2"/><i><font color="Green">E79</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="topreal9.html#T20">TOPREAL9:20</a></i>;<br/><a NAME="E6:38_1_2"/>
for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> holds <br/> <a href="metric_1.html#K9">Ball</a> <font color="Maroon">ae</font>,<font color="Olive">s</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:38_1_2_1"/><i><font color="Green">E80</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E2:38_1_2_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2_1_1"><b>now </b></a><div class="add"><a NAME="E1:38_1_2_1_1"/><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><a href="topreal9.html#K1">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal9.html#K3">Sphere</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal9.html#K2">cl_Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T18">TOPREAL9:18</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_1_2_1_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> )
 </span><b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:38_1_2_1_1_1_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 ;<br/><div class="add"><a NAME="E2:38_1_2_1_1_1_1"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K20">-</a> <font color="Maroon">a</font></span>)</span></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#T29">TOPRNS_1:29</a></i>;<br/><a NAME="E3:38_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">a</font>,<font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="topreal9.html#T7">TOPREAL9:7</a></i>;<br/><b>hence </b><a NAME="E4:38_1_2_1_1_1_1"/>
 <a href="topreal9.html#K1">Ball</a> <font color="Maroon">a</font>,<font color="Maroon">s</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:38_1_2_1_1_1_2"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 ;<br/><div class="add"><a NAME="E2:38_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E110</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</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="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T9">TOPREAL9:9</a></i>;<br/><a NAME="E3:38_1_2_1_1_1_2"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</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="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="toprns_1.html#T29">TOPRNS_1:29</a></i>;<br/><a NAME="E4:38_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="topreal9.html#T7">TOPREAL9:7</a></i>;<br/><b>set </b><font color="Maroon">z</font> = <font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span>;<br/><a NAME="E5:38_1_2_1_1_1_2"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> is   <a href="real_1.html#NM1">Real</a>
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><b>set </b><font color="Maroon">q</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">a</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span>;<br/><a NAME="E6:38_1_2_1_1_1_2"/>
1 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E7:38_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E8:38_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E118</font></i>: 
<font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T191">XREAL_1:191</a></i>;<br/><a NAME="E9:38_1_2_1_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T131">XREAL_1:131</a></i>;<br/><a NAME="E10:38_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E119</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T141">XREAL_1:141</a></i>;<br/><a NAME="E11:38_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E120</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">a</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">a</font>,<font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="sppol_1.html#T22">SPPOL_1:22</a></i>;<br/><a NAME="E12:38_1_2_1_1_1_2"/>
 <a href="topreal9.html#K1">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T19">TOPREAL9:19</a></i>;<br/><a NAME="E13:38_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E14:38_1_2_1_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">a</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">a</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> )
 <b>by </b><i>, , <a class="ref" href="topreal9.html#T4">TOPREAL9:4</a></i>;<br/><a NAME="E15:38_1_2_1_1_1_2"/><b>then </b>
not <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">a</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E16:38_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E121</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">a</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">a</font>,<font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E17:38_1_2_1_1_1_2"/><i><font color="Green">E122</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">a</font>,<font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font></span><a href="domain_1.html#K7">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">a</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">a</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">a</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="jordan2c.html#T9">JORDAN2C:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="euclid.html#K18">*</a> <font color="Maroon">a</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">a</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T54">EUCLID:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p5">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">a</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">x</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"><font color="Maroon">a</font> <a href="euclid.html#K20">-</a> <font color="Maroon">a</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="topreal9.html#T1">TOPREAL9:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T46">EUCLID:46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="jordan2c.html#T13">JORDAN2C:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K20">-</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T53">EUCLID:53</a></i>
;<br/><b>then </b><span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p0">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">a</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p0">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">a</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K20">-</a> <font color="Maroon">a</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i>, <a class="ref" href="toprns_1.html#T8">TOPRNS_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K20">-</a> <font color="Maroon">a</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="toprns_1.html#T28">TOPRNS_1:28</a></i>
<br/>.= 
<font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> 2
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T93">XCMPLX_1:93</a></i>
;<br/><a NAME="E20:38_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E123</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">a</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="topreal9.html#T9">TOPREAL9:9</a></i>;<br/><a NAME="E21:38_1_2_1_1_1_2"/>
<font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T78">XREAL_1:78</a></i>;<br/><a NAME="E22:38_1_2_1_1_1_2"/><b>then </b>
 <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">a</font>,<font color="Maroon">s</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E23:38_1_2_1_1_1_2"/>
 <a href="topreal9.html#K1">Ball</a> <font color="Maroon">a</font>,<font color="Maroon">s</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:38_1_2_1"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">ae</font>,<font color="Maroon">s</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="topreal9.html#K1">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T13">TOPREAL9:13</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:38_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="topreal9.html#K1">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="goboard6.html#T96">GOBOARD6:96</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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