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<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">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">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>assume </b><a NAME="E1:35"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">y</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1"/>
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1"><b>suppose </b></a><a NAME="E1:35_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">xe</font> = <font color="Maroon">x</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="E3:35_1_1"/>
 <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K11">Sphere</a> <font color="Maroon">xe</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="topreal9.html#T15">TOPREAL9:15</a></i>;<br/><a NAME="E4:35_1_1"/><b>then </b>
 <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="topreal6.html#T62">TOPREAL6:62</a></i>;<br/><a NAME="E5:35_1_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">x</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/><a NAME="E6:35_1_1"/><i><font color="Green">E78</font></i>: 
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">x</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="topreal1.html#T7">TOPREAL1:7</a></i>;<br/><a NAME="E7:35_1_1"/><i><font color="Green">E79</font></i>: 
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="domain_1.html#K7">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#T69">ENUMSET1:69</a></i>;<br/><a NAME="E8:35_1_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><b>hence </b><a NAME="E9:35_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">x</font>,<font color="Maroon">y</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">x</font>,<font color="Maroon">y</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>, , , <a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_2"><b>suppose </b></a><a NAME="E1:35_1_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>let </b><font color="Maroon">k</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="E2:35_1_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">x</font>,<font color="Maroon">y</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">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>
 ;<br/><a NAME="E3:35_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">l</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:35_1_2"/><i><font color="Green">E82</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>and </b><br/><a NAME="E6:35_1_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i><a class="ref" href="sppol_1.html#T21">SPPOL_1:21</a></i>;<br/><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</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="E8:35_1_2"/>
not <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</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="E9:35_1_2"/><b>then </b>
( <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E10:35_1_2"/><b>then </b>
( 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 <b>by </b><i>, <a class="ref" href="topreal9.html#T4">TOPREAL9:4</a></i>;<br/><a NAME="E11:35_1_2"/><b>then </b><i><font color="Green">E109</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;</a> 1 )
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><font color="Maroon">k</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font> = 
<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> <font color="Maroon">l</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">y</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">x</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">y</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">x</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">y</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">x</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">y</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"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">y</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"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">y</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"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">y</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID:45</a></i>
<br/>.= 
<font color="Maroon">l</font> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T53">EUCLID:53</a></i>
;<br/><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">k</font> <a href="euclid.html#K20">-</a> <font color="Maroon">x</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> <font color="Maroon">l</font></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">y</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#T8">TOPRNS_1:8</a></i>
<br/>.= 
<font color="Maroon">l</font> <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">y</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="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
<font color="Maroon">l</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font>
<b>by </b><i>, <a class="ref" href="topreal9.html#T9">TOPREAL9:9</a></i>
;<br/><a NAME="E14:35_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="topreal9.html#T39">TOPREAL9:39</a></i>;<br/><a NAME="E15:35_1_2"/><b>then </b>
<font color="Maroon">l</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <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/><b>hence </b><a NAME="E16:35_1_2"/>
<font color="Maroon">k</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/></div><b>end;</b></div></div><b>end;</b></div>

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