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

<b>let </b><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">q</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#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:4"/><i><font color="Green" title="E2">A1</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">q</font>
 ;<br/>

<a NAME="E2:4"/>
for <font color="Olive" title="b1">G</font> being  <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>  st <font color="Olive" title="b1">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">G</font> holds <br/><span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">G</font> be   <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>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:4_1"/><i><font color="Green" title="E3">A2</font></i>: 
<font color="Maroon" title="c3">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E2:4_1"/><i><font color="Green" title="E4">A3</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">P</font> = <font color="Maroon" title="c3">G</font> 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#K4" title="PCOMPS_1:func.4">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/>
<b>reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c6">y</font> = <font color="Maroon" title="c2">q</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#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13" title="TOPREAL3:th.13">TOPREAL3:13</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">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> <b> such that </b><br/><a NAME="E6:4_1"/><i><font color="Green" title="E5">A4</font></i>: 
<font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>and </b><br/><a NAME="E7:4_1"/><i><font color="Green" title="E6">A5</font></i>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">P</font>
 <b>by </b><i><a class="txt" href="jordan1h.html#E1:4_1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="txt" href="jordan1h.html#E2:4_1"><i><font color="Green" title="E4">A3</font></i></a>, <a class="ref" href="topmetr.html#T22" title="TOPMETR:th.22">TOPMETR:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">r</font> = <font color="Maroon" title="c7">r</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">t</font> =  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c10">s</font> = <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span>;<br/>
<a NAME="E9:4_1"/><i><font color="Green" title="E7">A6</font></i>: 
<font color="Maroon" title="c8">r</font> <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> 0
 
<b>by </b><i><a class="txt" href="jordan1h.html#E6:4_1"><i><font color="Green" title="E5">A4</font></i></a>, <a class="ref" href="real_2.html#T127" title="REAL_2:th.127">REAL_2:127</a></i>;<br/>
<a NAME="E10:4_1"/><i><font color="Green" title="E8">A7</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>
 
<b>by </b><i><a class="txt" href="jordan1h.html#E1:4"><i><font color="Green" title="E2">A1</font></i></a>, <a class="ref" href="sub_metr.html#T2" title="SUB_METR:th.2">SUB_METR:2</a></i>;<br/>
<a NAME="E11:4_1"/><b>then </b>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 
<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="E12:4_1"/><b>then </b><i><font color="Green" title="E9">A8</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1h.html#E9:4_1"><i><font color="Green" title="E7">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T21" title="XXREAL_0:th.21">XXREAL_0:21</a></i>;<br/>
<a NAME="E13:4_1"/><b>then </b><i><font color="Green" title="E10">A9</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1h.html#E10:4_1"><i><font color="Green" title="E8">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="E14:4_1"/><i><font color="Green" title="E11">A10</font></i>: 
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E15:4_1"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 1
 
<b>by </b><i><a class="txt" href="jordan1h.html#E10:4_1"><i><font color="Green" title="E8">A7</font></i></a>, <a class="ref" href="real_2.html#T200" title="REAL_2:th.200">REAL_2:200</a></i>;<br/>
<a NAME="E16:4_1"/><b>then </b><i><font color="Green" title="E12">A11</font></i>: 
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>
 
<b>by </b><i><a class="txt" href="jordan1h.html#E14:4_1"><i><font color="Green" title="E11">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E17:4_1"/><b>then </b><i><font color="Green" title="E13">A12</font></i>: 
<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 
<b>by </b><i><a class="txt" href="jordan1h.html#E12:4_1"><i><font color="Green" title="E9">A8</font></i></a>, <a class="ref" href="real_2.html#T142" title="REAL_2:th.142">REAL_2:142</a></i>;<br/>
<b>set </b><font color="Maroon" title="c11">pp</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">q</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">z</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">q</font></span>)</span> 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#K14" title="EUCLID:func.14">Euclid</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="E19:4_1"/><i><font color="Green" title="E14">A13</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font>
 
<b>by </b><i><a class="txt" href="jordan1h.html#E13:4_1"><i><font color="Green" title="E10">A9</font></i></a>, <a class="txt" href="jordan1h.html#E17:4_1"><i><font color="Green" title="E13">A12</font></i></a>, <a class="ref" href="sppol_1.html#T22" title="SPPOL_1:th.22">SPPOL_1:22</a></i>;<br/>
<i><font color="Green" title="E15">A14</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c1">p</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c1">p</font>
<b>by </b><i><a class="ref" href="euclid.html#T37" title="EUCLID:th.37">EUCLID:37</a></i>
<br/>.= 
<font color="Maroon" title="c1">p</font>
<b>by </b><i><a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>
;<br/>
<div><i><font color="Green" title="E16">A15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:4_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font>
 ;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c1">p</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" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">q</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c1">p</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan1h.html#E20:4_1"><i><font color="Green" title="E15">A14</font></i></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">q</font>
<b>by </b><i><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></i>
;<br/><b>hence </b><a NAME="E3:4_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan1h.html#E1:4"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="jordan1h.html#E13:4_1"><i><font color="Green" title="E10">A9</font></i></a>, <a class="ref" href="euclid.html#T38" title="EUCLID:th.38">EUCLID:38</a></i>;<br/></div><b>end;</b></div>
<i><font color="Green" title="E17">A16</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">q</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">q</font>
<b>by </b><i><a class="ref" href="euclid.html#T37" title="EUCLID:th.37">EUCLID:37</a></i>
<br/>.= 
<font color="Maroon" title="c2">q</font>
<b>by </b><i><a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>
;<br/>
<a NAME="E23:4_1"/><i><font color="Green" title="E18">A17</font></i>: 
1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="jordan1h.html#E16:4_1"><i><font color="Green" title="E12">A11</font></i></a>, <a class="txt" href="jordan1h.html#E12:4_1"><i><font color="Green" title="E9">A8</font></i></a>, <a class="ref" href="real_2.html#T142" title="REAL_2:th.142">REAL_2:142</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:4_1_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">q</font>
 ;<br/><b>then </b><span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">q</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" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">q</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">q</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan1h.html#E22:4_1"><i><font color="Green" title="E17">A16</font></i></a>, <a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c1">p</font>
<b>by </b><i><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></i>
;<br/><b>hence </b><a NAME="E3:4_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan1h.html#E1:4"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="jordan1h.html#E23:4_1"><i><font color="Green" title="E18">A17</font></i></a>, <a class="ref" href="euclid.html#T38" title="EUCLID:th.38">EUCLID:38</a></i>;<br/></div><b>end;</b></div>
<a NAME="E25:4_1"/><b>then </b>
not <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="txt" href="jordan1h.html#E21:4_1"><i><font color="Green" title="E16">A15</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E26:4_1"/><b>then </b><i><font color="Green" title="E19">A18</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="txt" href="jordan1h.html#E19:4_1"><i><font color="Green" title="E14">A13</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">x'</font> = <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c14">y'</font> = <font color="Maroon" title="c6">y</font>, <font color="Maroon" title="c15">z'</font> = <font color="Maroon" title="c12">z</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2 ;<br/>
<b>reconsider </b><font color="Maroon" title="c16">a</font> = <font color="Maroon" title="c13">x'</font>, <font color="Maroon" title="c17">b</font> = <font color="Maroon" title="c14">y'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c18">u</font> = <font color="Maroon" title="c16">a</font> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <font color="Maroon" title="c17">b</font>, <font color="Maroon" title="c19">v</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c17">b</font>, <font color="Maroon" title="c20">w</font> = <span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c16">a</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2 ;<br/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c12">z</font> = 
<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="c13">x'</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c15">z'</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="sppol_1.html#T20" title="SPPOL_1:th.20">SPPOL_1:20</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c16">a</font></span>)</span></span>)</span> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c17">b</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T60" title="RVSUM_1:th.60">RVSUM_1:60</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c16">a</font></span>)</span></span>)</span> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c17">b</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T74" title="RVSUM_1:th.74">RVSUM_1:74</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K7" title="RVSUM_1:func.7">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c16">a</font></span>)</span></span>)</span></span>)</span> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c17">b</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T52" title="RVSUM_1:th.52">RVSUM_1:52</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c16">a</font></span>)</span></span>)</span></span>)</span> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c17">b</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T76" title="RVSUM_1:th.76">RVSUM_1:76</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c16">a</font></span>)</span></span>)</span> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c17">b</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T71" title="RVSUM_1:th.71">RVSUM_1:71</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p0">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c17">b</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T72" title="RVSUM_1:th.72">RVSUM_1:72</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</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"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K7" title="RVSUM_1:func.7">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p0">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c17">b</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T52" title="RVSUM_1:th.52">RVSUM_1:52</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</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"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p0">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c17">b</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T76" title="RVSUM_1:th.76">RVSUM_1:76</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</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"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p0">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c17">b</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T71" title="RVSUM_1:th.71">RVSUM_1:71</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</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"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c16">a</font></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p5">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c17">b</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T71" title="RVSUM_1:th.71">RVSUM_1:71</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c17">b</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T73" title="RVSUM_1:th.73">RVSUM_1:73</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K7" title="RVSUM_1:func.7">-</a> <font color="Maroon" title="c17">b</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T76" title="RVSUM_1:th.76">RVSUM_1:76</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">a</font> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <font color="Maroon" title="c17">b</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T52" title="RVSUM_1:th.52">RVSUM_1:52</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c18">u</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T14" title="EUCLID:th.14">EUCLID:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</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="c16">a</font> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <font color="Maroon" title="c17">b</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="txt" href="jordan1h.html#E13:4_1"><i><font color="Green" title="E10">A9</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span>
<b>by </b><i><a class="ref" href="sppol_1.html#T20" title="SPPOL_1:th.20">SPPOL_1:20</a></i>
<br/>.= 
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>
<b>by </b><i><a class="txt" href="jordan1h.html#E10:4_1"><i><font color="Green" title="E8">A7</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>
;<br/>
<a NAME="E31:4_1"/><b>then </b><i><font color="Green" title="E20">A19</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c12">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E32:4_1"/>
<font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 1
 
<b>by </b><i><a class="txt" href="jordan1h.html#E6:4_1"><i><font color="Green" title="E5">A4</font></i></a>, <a class="ref" href="real_2.html#T200" title="REAL_2:th.200">REAL_2:200</a></i>;<br/>
<a NAME="E33:4_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c12">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">r</font>
 
<b>by </b><i><a class="txt" href="jordan1h.html#E31:4_1"><i><font color="Green" title="E20">A19</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E34:4_1"/><b>then </b>
<font color="Maroon" title="c12">z</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="c5">x</font>,<font color="Maroon" title="c8">r</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<b>hence </b><a NAME="E35:4_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">G</font>
 <b>by </b><i><a class="txt" href="jordan1h.html#E7:4_1"><i><font color="Green" title="E6">A5</font></i></a>, <a class="txt" href="jordan1h.html#E26:4_1"><i><font color="Green" title="E19">A18</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:4"/>
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="tops_1.html#T39" title="TOPS_1:th.39">TOPS_1:39</a></i>;<br/>


</div>
