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<div class="add">

<span class="kw">defpred </span><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>] <span class="kw">means </span><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/>
<span class="kw">reconsider </span><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> <span class="kw">from</span> <span class="lab"><a class="ref" href="jgraph_2.html#S1" target="_self" title="JGRAPH_2:sch.1">JGRAPH_2:sch 1</a>();<br/></span>
<a NAME="E2:40_1"/><span class="lab"><font color="Green" title="E42">A3</font></span>: 
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#D8" title="EUCLID:def.8">EUCLID:def 8</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">K21</font> = <font color="Maroon" title="c1">K2</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span></span>)</span> ;<br/>
<a NAME="E4:40_1"/><span class="lab"><font color="Green" title="E43">A4</font></span>: 
<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="E5:40_1"/><span class="lab"><font color="Green" title="E44">A5</font></span>: 
<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> 
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="jgraph_2.html#S2" target="_self" title="JGRAPH_2:sch.2">JGRAPH_2:sch 2</a>(<a class="txt" href="#E4:40_1"><span class="lab"><font color="Green" title="E43">A4</font></span></a>);<br/></span>
<a NAME="E6:40_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="c2">K21</font> 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="c2">K21</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K21</font> implies  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="c3">p</font>,<font color="Olive" title="b1">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">K21</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:40_1_1"/><span class="lab"><font color="Green" title="E45">A6</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K21</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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="c3">p</font>,<font color="Olive" title="b1">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">K21</font> )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">p0</font> = <font color="Maroon" title="c3">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/>
<span class="kw">set </span><font color="Maroon" title="c5">r0</font> = <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c4">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="c4">p0</font>)</span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4;<br/>
<a NAME="E3:40_1_1"/>
 ex <font color="Olive" title="b1">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> st <br/>( <font color="Maroon" title="c4">p0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p7</font> &amp; <font color="Maroon">F<sub>1</sub></font>(<font color="Olive" title="b1">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="Olive" title="b1">p7</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_1"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="txt" href="#E1:40_1_1"><span class="lab"><font color="Green" title="E45">A6</font></span></a></span>;<br/>
<a NAME="E4:40_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A7</font></span>: 
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c4">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="c4">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/>
<a NAME="E5:40_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A8</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c4">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="c4">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T139" title="XREAL_1:th.139">XREAL_1:139</a></span>;<br/>
<a NAME="E6:40_1_1"/>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c3">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="c4">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="c4">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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <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="c3">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="c4">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="c4">p0</font>)</span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4</span>)</span>) or <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>  )</span><br/>

<a NAME="E1:40_1_1_1"/><span class="lab"><font color="Green" title="E48">A9</font></span>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c3">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="c4">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="c4">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="c3">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="c4">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="c4">p0</font>)</span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4 </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T17" title="METRIC_1:th.17">METRIC_1:17</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:40_1_1_1"/><span class="lab"><font color="Green" title="E49">A10</font></span>: 
<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="c3">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="c4">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="c4">p0</font>)</span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4</span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal3.html#T8" title="TOPREAL3:th.8">TOPREAL3:8</a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E5:40_1_1_1"/><span class="lab"><font color="Green" title="E50">A11</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 <span class="kw">and </span><br/><a NAME="E6:40_1_1_1"/><span class="lab"><font color="Green" title="E51">A12</font></span>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c3">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="c4">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="c4">p0</font>)</span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_1_1"><span class="lab"><font color="Green" title="E49">A10</font></span></a>, <a class="txt" href="#E1:40_1_1_1"><span class="lab"><font color="Green" title="E48">A9</font></span></a></span>;<br/>
<a NAME="E7:40_1_1_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c3">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="c4">p0</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c7">px</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_1_1_1"><span class="lab"><font color="Green" title="E50">A11</font></span></a>, <a class="ref" href="jgraph_1.html#T28" title="JGRAPH_1:th.28">JGRAPH_1:28</a></span>;<br/>
<a NAME="E8:40_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A13</font></span>: 
<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="c4">p0</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</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="c4">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="c4">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40_1_1_1"><span class="lab"><font color="Green" title="E51">A12</font></span></a>, <a class="ref" href="square_1.html#T15" title="SQUARE_1:th.15">SQUARE_1:15</a></span>;<br/>
<a NAME="E9:40_1_1_1"/><span class="lab"><font color="Green" title="E53">A14</font></span>: 
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">p0</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</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="c4">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E40">A1</font></span></a></span>;<br/>
<a NAME="E10:40_1_1_1"/><span class="lab"><font color="Green" title="E54">A15</font></span>: 
<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="c4">p0</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</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="c4">p0</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</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="c4">p0</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E41">A2</font></span></a></span>;<br/>
<a NAME="E11:40_1_1_1"/>
<font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">p0</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T63" title="XREAL_1:th.63">XREAL_1:63</a></span>;<br/>
<a NAME="E12:40_1_1_1"/><span class="kw">then </span>
<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="c4">p0</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</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="c4">p0</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</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="c4">p0</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<a NAME="E13:40_1_1_1"/><span class="kw">then </span>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">p0</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</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="c4">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="c4">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:40_1_1_1"><span class="lab"><font color="Green" title="E54">A15</font></span></a>, <a class="txt" href="#E8:40_1_1_1"><span class="lab"><font color="Green" title="E52">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E14:40_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A16</font></span>: 
<font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c4">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="c4">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="c4">p0</font>)</span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1_1"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="txt" href="#E9:40_1_1_1"><span class="lab"><font color="Green" title="E53">A14</font></span></a>, <a class="ref" href="square_1.html#T47" title="SQUARE_1:th.47">SQUARE_1:47</a></span>;<br/>
<a NAME="E15:40_1_1_1"/><span class="lab"><font color="Green" title="E56">A17</font></span>: 
<font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">p0</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</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="c4">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E40">A1</font></span></a></span>;<br/>
<a NAME="E16:40_1_1_1"/>
<font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">p0</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T63" title="XREAL_1:th.63">XREAL_1:63</a></span>;<br/>
<a NAME="E17:40_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">p0</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</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="c4">p0</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</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="c4">p0</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<a NAME="E18:40_1_1_1"/><span class="kw">then </span>
<font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c4">p0</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</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="c4">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="c4">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:40_1_1_1"><span class="lab"><font color="Green" title="E54">A15</font></span></a>, <a class="txt" href="#E8:40_1_1_1"><span class="lab"><font color="Green" title="E52">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E19:40_1_1_1"/><span class="kw">then </span>
 <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="c4">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="c4">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="c4">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1_1"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="txt" href="#E15:40_1_1_1"><span class="lab"><font color="Green" title="E56">A17</font></span></a>, <a class="ref" href="square_1.html#T47" title="SQUARE_1:th.47">SQUARE_1:47</a></span>;<br/>
<a NAME="E20:40_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>(<font color="Maroon" title="c4">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="c4">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="c4">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="c4">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="c4">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="c4">p0</font>)</span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 4</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:40_1_1_1"><span class="lab"><font color="Green" title="E55">A16</font></span></a>, <a class="ref" href="xreal_1.html#T13" title="XREAL_1:th.13">XREAL_1:13</a></span>;<br/>
<a NAME="E21:40_1_1_1"/><span class="kw">then </span>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_1_1"><span class="lab"><font color="Green" title="E47">A8</font></span></a>, <a class="txt" href="jgraph_2.html#E40"><span class="lab"><font color="Green" title="E39">Lm2</font></span></a></span>;<br/>
<a NAME="E22:40_1_1_1"/><span class="kw">then </span>
<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>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T47" title="XREAL_1:th.47">XREAL_1:47</a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:40_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_1"><span class="lab"><font color="Green" title="E44">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:40_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="c3">p</font>,<font color="Olive" title="b1">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">K21</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1_1"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="ref" href="xreal_1.html#T139" title="XREAL_1:th.139">XREAL_1:139</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:40_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">K21</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T15" title="TOPMETR:th.15">TOPMETR:15</a></span>;<br/>
<a NAME="E8:40_1"/><span class="kw">then </span>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="ref" href="pre_topc.html#T30" title="PRE_TOPC:th.30">PRE_TOPC:30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:40_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T3" title="TOPS_1:th.3">TOPS_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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