<?xml version="1.0"?>
<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>] <b>means </b><font color="Maroon">F<sub>1</sub></font>($1) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>2</sub></font>($1);<br/>
<b>reconsider </b><font color="Maroon" title="c1">K2</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">p7</font> where <font color="Olive" title="b1">p7</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p7</font>] </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>from </b><i><a class="ref" href="jgraph_2.html#S1" target="_self" title="JGRAPH_2:sch.1">JGRAPH_2:sch 1</a>();<br/></i>
<a NAME="E2:40_1"/><i><font color="Green" title="E41">A3</font></i>: 
<font color="Maroon" title="c1">K2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p7</font> where <font color="Olive" title="b1">p7</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p7</font>] </span>}</span> 
 
;<br/>
<a NAME="E3:40_1"/><i><font color="Green" title="E42">A4</font></i>: 
<font color="Maroon" title="c1">K2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p7</font> where <font color="Olive" title="b1">p7</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> :  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p7</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="jgraph_2.html#S2" target="_self" title="JGRAPH_2:sch.2">JGRAPH_2:sch 2</a>(<a class="txt" href="jgraph_2.html#E2:40_1"><i><font color="Green" title="E41">A3</font></i></a>);<br/></i>
<a NAME="E4:40_1"/>
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">K2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  holds <br/> ex <font color="Olive" title="b2">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b2">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">K2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:40_1_1"/><i><font color="Green" title="E43">A5</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">K2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c3">p0</font> = <font color="Maroon" title="c2">p</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>set </b><font color="Maroon" title="c4">r0</font> = <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">p0</font>) <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">p0</font>)</span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4;<br/>
<b>consider </b><font color="Maroon" title="c5">p7</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E4:40_1_1"/><i><font color="Green" title="E44">A6</font></i>: 
( <font color="Maroon" title="c3">p0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p7</font> &amp; <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c5">p7</font>) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c5">p7</font>) )
 <b>by </b><i><a class="txt" href="jgraph_2.html#E3:40_1"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="jgraph_2.html#E1:40_1_1"><i><font color="Green" title="E43">A5</font></i></a></i>;<br/>
<a NAME="E5:40_1_1"/><i><font color="Green" title="E45">A7</font></i>: 
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">p0</font>) <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">p0</font>) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E4:40_1_1"><i><font color="Green" title="E44">A6</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/>
<a NAME="E6:40_1_1"/><b>then </b><i><font color="Green" title="E46">A8</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">p0</font>) <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">p0</font>)</span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="real_2.html#T127" title="REAL_2:th.127">REAL_2:127</a></i>;<br/>
<a NAME="E7:40_1_1"/><i><font color="Green" title="E47">A9</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">p0</font>) <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">p0</font>)</span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E5:40_1_1"><i><font color="Green" title="E45">A7</font></i></a>, <a class="ref" href="real_2.html#T127" title="REAL_2:th.127">REAL_2:127</a></i>;<br/>
<a NAME="E8:40_1_1"/>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">p0</font>) <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">p0</font>)</span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">K2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:40_1_1_1"/><i><font color="Green" title="E48">A10</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">p0</font>) <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">p0</font>)</span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4</span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c7">px</font> = <font color="Maroon" title="c6">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13" title="TOPREAL3:th.13">TOPREAL3:13</a></i>;<br/>
<a NAME="E3:40_1_1_1"/>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">p0</font>) <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">p0</font>)</span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> :  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">q</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">p0</font>) <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">p0</font>)</span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4 </span>}</span> 
 
<b>by </b><i><a class="ref" href="metric_1.html#T18" title="METRIC_1:th.18">METRIC_1:18</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span><b> such that </b><br/><a NAME="E5:40_1_1_1"/><i><font color="Green" title="E49">A11</font></i>: 
( <font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> &amp;  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c8">q</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">p0</font>) <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">p0</font>)</span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4 )
 <b>by </b><i><a class="txt" href="jgraph_2.html#E1:40_1_1_1"><i><font color="Green" title="E48">A10</font></i></a></i>;<br/>
<a NAME="E6:40_1_1_1"/><i><font color="Green" title="E50">A12</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p0</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">px</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E5:40_1_1_1"><i><font color="Green" title="E49">A11</font></i></a>, <a class="ref" href="jgraph_1.html#T45" title="JGRAPH_1:th.45">JGRAPH_1:45</a></i>;<br/>
<a NAME="E7:40_1_1_1"/><i><font color="Green" title="E51">A13</font></i>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p0</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">px</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c3">p0</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">px</font></span>)</span>) <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c3">p0</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">px</font></span>)</span>) <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E2:40"><i><font color="Green" title="E40">A2</font></i></a></i>;<br/>
<a NAME="E8:40_1_1_1"/><i><font color="Green" title="E52">A14</font></i>: 
<font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c3">p0</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">px</font></span>)</span>) <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></i>;<br/>
<a NAME="E9:40_1_1_1"/>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c3">p0</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">px</font></span>)</span>) <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></i>;<br/>
<a NAME="E10:40_1_1_1"/><b>then </b><i><font color="Green" title="E53">A15</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c3">p0</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">px</font></span>)</span>) <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c3">p0</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">px</font></span>)</span>) <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c3">p0</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">px</font></span>)</span>) <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E11:40_1_1_1"/><i><font color="Green" title="E54">A16</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c3">p0</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">px</font></span>)</span>) <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c3">p0</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">px</font></span>)</span>) <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c3">p0</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">px</font></span>)</span>) <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E8:40_1_1_1"><i><font color="Green" title="E52">A14</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E12:40_1_1_1"/><i><font color="Green" title="E55">A17</font></i>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p0</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">px</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">p0</font>) <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">p0</font>)</span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E5:40_1_1_1"><i><font color="Green" title="E49">A11</font></i></a>, <a class="txt" href="jgraph_2.html#E6:40_1_1_1"><i><font color="Green" title="E50">A12</font></i></a>, <a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></i>;<br/>
<a NAME="E13:40_1_1_1"/><b>then </b><i><font color="Green" title="E56">A18</font></i>: 
<font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c3">p0</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">px</font></span>)</span>) <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">p0</font>) <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">p0</font>)</span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E7:40_1_1_1"><i><font color="Green" title="E51">A13</font></i></a>, <a class="txt" href="jgraph_2.html#E10:40_1_1_1"><i><font color="Green" title="E53">A15</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E14:40_1_1_1"/><i><font color="Green" title="E57">A19</font></i>: 
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c3">p0</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">px</font></span>)</span>) <a href="square_1.html#K3" title="SQUARE_1:func.3">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">p0</font>) <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">p0</font>)</span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E7:40_1_1_1"><i><font color="Green" title="E51">A13</font></i></a>, <a class="txt" href="jgraph_2.html#E11:40_1_1_1"><i><font color="Green" title="E54">A16</font></i></a>, <a class="txt" href="jgraph_2.html#E12:40_1_1_1"><i><font color="Green" title="E55">A17</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E15:40_1_1_1"/><i><font color="Green" title="E58">A20</font></i>: 
( <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c3">p0</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">px</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">p0</font>) <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c7">px</font>) &amp; <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c3">p0</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <font color="Maroon" title="c7">px</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">p0</font>) <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c7">px</font>) )
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E1:40"><i><font color="Green" title="E39">A1</font></i></a></i>;<br/>
<a NAME="E16:40_1_1_1"/><b>then </b><i><font color="Green" title="E59">A21</font></i>: 
(  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">p0</font>) <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">p0</font>)</span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">p0</font>) <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c7">px</font>) &amp; <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">p0</font>) <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c7">px</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">p0</font>) <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">p0</font>)</span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4 )
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E6:40_1_1"><i><font color="Green" title="E46">A8</font></i></a>, <a class="txt" href="jgraph_2.html#E13:40_1_1_1"><i><font color="Green" title="E56">A18</font></i></a>, <a class="ref" href="square_1.html#T117" title="SQUARE_1:th.117">SQUARE_1:117</a></i>;<br/>
<a NAME="E17:40_1_1_1"/>
(  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">p0</font>) <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">p0</font>)</span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">p0</font>) <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c7">px</font>) &amp; <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">p0</font>) <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c7">px</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">p0</font>) <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">p0</font>)</span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4 )
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E6:40_1_1"><i><font color="Green" title="E46">A8</font></i></a>, <a class="txt" href="jgraph_2.html#E14:40_1_1_1"><i><font color="Green" title="E57">A19</font></i></a>, <a class="txt" href="jgraph_2.html#E15:40_1_1_1"><i><font color="Green" title="E58">A20</font></i></a>, <a class="ref" href="square_1.html#T117" title="SQUARE_1:th.117">SQUARE_1:117</a></i>;<br/>
<a NAME="E18:40_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">p0</font>) <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c7">px</font>)</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">p0</font>) <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c7">px</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">p0</font>) <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">p0</font>)</span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c3">p0</font>) <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">p0</font>)</span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E16:40_1_1_1"><i><font color="Green" title="E59">A21</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/>
<a NAME="E19:40_1_1_1"/><b>then </b>
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c7">px</font>) <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c7">px</font>) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jgraph_2.html#E7:40_1_1"><i><font color="Green" title="E47">A9</font></i></a>, <a class="txt" href="jgraph_2.html#E49"><i><font color="Green" title="E38">Lm1</font></i></a></i>;<br/>
<a NAME="E20:40_1_1_1"/><b>then </b>
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c7">px</font>) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c7">px</font>)
 
<b>by </b><i><a class="ref" href="real_2.html#T106" title="REAL_2:th.106">REAL_2:106</a></i>;<br/>
<b>hence </b><a NAME="E21:40_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">K2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="jgraph_2.html#E3:40_1"><i><font color="Green" title="E42">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:40_1_1"/>
 ex <font color="Olive" title="b1">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">K2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 <b>by </b><i><a class="txt" href="jgraph_2.html#E6:40_1_1"><i><font color="Green" title="E46">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:40_1"/><b>then </b>
<font color="Maroon" title="c1">K2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="topmetr.html#T22" title="TOPMETR:th.22">TOPMETR:22</a></i>;<br/>
<b>hence </b><a NAME="E6:40_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">p</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">p</font>) </span>}</span>  is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="tops_1.html#T29" title="TOPS_1:th.29">TOPS_1:29</a></i>;<br/>


</div>
