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

<b>let </b><font color="Maroon" title="c1">P7</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a> ;<br/>

<b>assume </b><a NAME="E1:41"/><i><font color="Green" title="E26">A1</font></i>: 
<font color="Maroon" title="c1">P7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ;<br/>

<a NAME="E2:41"/><i><font color="Green" title="E27">A2</font></i>: 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E3:41"/><i><font color="Green" title="E28">A3</font></i>: 
not 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E4:41"/><b>then </b><i><font color="Green" title="E29">A4</font></i>: 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="txt" href="jordan6.html#E2:41"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E5:41"/><i><font color="Green" title="E30">A5</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P7</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P7</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E6:41"/>
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P7</font></span>)</span>  st  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P7</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P7</font></span>)</span> &amp; <font color="Olive" title="b2">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P7</font></span>)</span> &amp; <font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/><font color="Olive" title="b1">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b2">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P7</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:41_1"/><i><font color="Green" title="E31">A6</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P7</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font>
 <b>and </b><br/><a NAME="E2:41_1"/><i><font color="Green" title="E32">A7</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P7</font></span>)</span> &amp; <font color="Maroon" title="c3">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P7</font></span>)</span> )
 <b>and </b><br/><a NAME="E3:41_1"/><i><font color="Green" title="E33">A8</font></i>: 
<font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E4:41_1"/><i><font color="Green" title="E34">A9</font></i>: 
<font color="Maroon" title="c3">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<b>assume </b><a NAME="E5:41_1"/><i><font color="Green" title="E35">A10</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">B</font>
 ;<br/>

<a NAME="E6:41_1"/><i><font color="Green" title="E36">A11</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T46" title="RCOMP_1:th.46">RCOMP_1:46</a></i>;<br/>
<i><font color="Green" title="E37">A12</font></i>: <font color="Maroon" title="c1">P7</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<b>by </b><i><a class="txt" href="jordan6.html#E1:41"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T11" title="RCOMP_1:th.11">RCOMP_1:11</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T40" title="XBOOLE_1:th.40">XBOOLE_1:40</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
<b>by </b><i><a class="txt" href="jordan6.html#E6:41_1"><i><font color="Green" title="E36">A11</font></i></a>, <a class="ref" href="xboole_1.html#T83" title="XBOOLE_1:th.83">XBOOLE_1:83</a></i>
;<br/>
<b>reconsider </b><font color="Maroon" title="c4">D1</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">P1</font> = <font color="Maroon" title="c1">P7</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="txt" href="jordan6.html#E7:41_1"><i><font color="Green" title="E37">A12</font></i></a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E10:41_1"/>
 <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">D1</font>
 
<b>by </b><i><a class="ref" href="topmetr.html#T26" title="TOPMETR:th.26">TOPMETR:26</a>, <a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a></i>;<br/>
<a NAME="E11:41_1"/><b>then </b><i><font color="Green" title="E38">A13</font></i>: 
<a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P7</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c5">P1</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="pre_topc.html#T28" title="PRE_TOPC:th.28">PRE_TOPC:28</a></i>;<br/>
<a NAME="E12:41_1"/>
<font color="Maroon" title="c5">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r1</font> where <font color="Olive" title="b1">r1</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">r1</font> &amp; <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan6.html#E7:41_1"><i><font color="Green" title="E37">A12</font></i></a>, <a class="ref" href="rcomp_1.html#D2" title="RCOMP_1:def.2">RCOMP_1:def 2</a></i>;<br/>
<a NAME="E13:41_1"/><b>then </b>
<font color="Maroon" title="c5">P1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="jordan2b.html#T32" title="JORDAN2B:th.32">JORDAN2B:32</a></i>;<br/>
<a NAME="E14:41_1"/><b>then </b><i><font color="Green" title="E39">A14</font></i>: 
<a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P7</font> is  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of  <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> 
 
<b>by </b><i><a class="txt" href="jordan6.html#E1:41"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="jordan6.html#E4:41"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="jordan6.html#E5:41"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="jordan6.html#E11:41_1"><i><font color="Green" title="E38">A13</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a>, <a class="ref" href="tsep_1.html#T16" title="TSEP_1:th.16">TSEP_1:16</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">P</font> = <font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c7">Q</font> = <font color="Maroon" title="c3">B</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="txt" href="jordan6.html#E5:41"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="jordan6.html#E2:41_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="txt" href="jordan6.html#E7:41_1"><i><font color="Green" title="E37">A12</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">A0</font> = <font color="Maroon" title="c6">P</font>, <font color="Maroon" title="c9">B0</font> = <font color="Maroon" title="c7">Q</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a>, <a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a>, <a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">TOPMETR:def 7</a></i>;<br/>
<a NAME="E17:41_1"/><i><font color="Green" title="E40">A15</font></i>: 
( <font color="Maroon" title="c8">A0</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c9">B0</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 
<b>by </b><i><a class="txt" href="jordan6.html#E3:41_1"><i><font color="Green" title="E33">A8</font></i></a>, <a class="txt" href="jordan6.html#E4:41_1"><i><font color="Green" title="E34">A9</font></i></a>, <a class="txt" href="jordan6.html#E14:41_1"><i><font color="Green" title="E39">A14</font></i></a>, <a class="ref" href="tsep_1.html#T17" title="TSEP_1:th.17">TSEP_1:17</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">xp</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c6">P</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">xp</font> = <font color="Maroon" title="c10">xp</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E20:41_1"/><i><font color="Green" title="E41">A16</font></i>: 
<font color="Maroon" title="c11">xp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 
;<br/>
<a NAME="E21:41_1"/><i><font color="Green" title="E42">A17</font></i>: 
<font color="Maroon" title="c6">P</font> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_2_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c12">p</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/><b>let </b><font color="Maroon" title="c13">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>assume </b><a NAME="E1:41_1_2_1"/>
<font color="Maroon" title="c13">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 ;<br/><a NAME="E2:41_1_2_1"/><b>then </b>
<font color="Maroon" title="c13">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="jordan6.html#E5:41"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="jordan6.html#E7:41_1"><i><font color="Green" title="E37">A12</font></i></a></i>;<br/><a NAME="E3:41_1_2_1"/><b>then </b>
<font color="Maroon" title="c13">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">w</font> &amp; <font color="Olive" title="b1">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D2" title="RCOMP_1:def.2">RCOMP_1:def 2</a></i>;<br/><a NAME="E4:41_1_2_1"/><b>then </b>
 ex <font color="Olive" title="b1">u</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">r</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 )
 ;<br/><b>hence </b><a NAME="E5:41_1_2_1"/>
<font color="Maroon" title="c12">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">r</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:41_1_2"/>
<font color="Maroon" title="c6">P</font> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a>
 <b>by </b><i><a class="ref" href="seq_4.html#D2" title="SEQ_4:def.2">SEQ_4:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:41_1"/><i><font color="Green" title="E43">A18</font></i>: 
<font color="Maroon" title="c7">Q</font> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_3_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c12">p</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/><b>let </b><font color="Maroon" title="c13">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>assume </b><a NAME="E1:41_1_3_1"/>
<font color="Maroon" title="c13">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Q</font>
 ;<br/><a NAME="E2:41_1_3_1"/><b>then </b>
<font color="Maroon" title="c13">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="jordan6.html#E5:41"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="jordan6.html#E7:41_1"><i><font color="Green" title="E37">A12</font></i></a></i>;<br/><a NAME="E3:41_1_3_1"/><b>then </b>
<font color="Maroon" title="c13">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">w</font> &amp; <font color="Olive" title="b1">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D2" title="RCOMP_1:def.2">RCOMP_1:def 2</a></i>;<br/><a NAME="E4:41_1_3_1"/><b>then </b>
 ex <font color="Olive" title="b1">u</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">r</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 )
 ;<br/><b>hence </b><a NAME="E5:41_1_3_1"/>
<font color="Maroon" title="c12">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">r</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:41_1_3"/>
<font color="Maroon" title="c7">Q</font> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a>
 <b>by </b><i><a class="ref" href="seq_4.html#D2" title="SEQ_4:def.2">SEQ_4:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c12">l</font> =  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c7">Q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c13">m</font> = <font color="Maroon" title="c12">l</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">t</font> = <font color="Maroon" title="c13">m</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a>, <a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">TOPMETR:def 7</a></i>;<br/>
<a NAME="E26:41_1"/><i><font color="Green" title="E44">A19</font></i>: 
not <font color="Maroon" title="c12">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1_4"/>
<font color="Maroon" title="c12">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Q</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c15">s</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="E3:41_1_4"/><i><font color="Green" title="E45">A20</font></i>: 
<font color="Maroon" title="c15">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E4:41_1_4"/><i><font color="Green" title="E46">A21</font></i>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c13">m</font>,<font color="Maroon" title="c15">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">B0</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E17:41_1"><i><font color="Green" title="E40">A15</font></i></a>, <a class="ref" href="topmetr.html#T22" title="TOPMETR:th.22">TOPMETR:22</a>, <a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">TOPMETR:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">s</font> = <font color="Maroon" title="c15">s</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.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>reconsider </b><font color="Maroon" title="c17">e1</font> = <font color="Maroon" title="c12">l</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">s</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E7:41_1_4"/><i><font color="Green" title="E47">A22</font></i>: 
<font color="Maroon" title="c16">s</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jordan6.html#E3:41_1_4"><i><font color="Green" title="E45">A20</font></i></a>, <a class="ref" href="real_2.html#T127" title="REAL_2:th.127">REAL_2:127</a></i>;<br/>
<a NAME="E8:41_1_4"/>
<font color="Maroon" title="c16">s</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c16">s</font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E3:41_1_4"><i><font color="Green" title="E45">A20</font></i></a>, <a class="ref" href="xreal_1.html#T218" title="XREAL_1:th.218">XREAL_1:218</a></i>;<br/>
<a NAME="E9:41_1_4"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">s</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c16">s</font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E7:41_1_4"><i><font color="Green" title="E47">A22</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E10:41_1_4"/><b>then </b>
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c13">m</font>,<font color="Maroon" title="c17">e1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c16">s</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D13" title="METRIC_1:def.13">METRIC_1:def 13</a>, <a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E11:41_1_4"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c13">m</font>,<font color="Maroon" title="c17">e1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c16">s</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/>
<a NAME="E12:41_1_4"/><b>then </b>
<font color="Maroon" title="c17">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  :  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c13">m</font>,<font color="Olive" title="b1">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c16">s</font> </span>}</span> 
 
;<br/>
<a NAME="E13:41_1_4"/><b>then </b>
<font color="Maroon" title="c12">l</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">s</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <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="c13">m</font>,<font color="Maroon" title="c16">s</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T18" title="METRIC_1:th.18">METRIC_1:18</a></i>;<br/>
<a NAME="E14:41_1_4"/><b>then </b>
<font color="Maroon" title="c12">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">l</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">s</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan6.html#E22:41_1"><i><font color="Green" title="E43">A18</font></i></a>, <a class="txt" href="jordan6.html#E4:41_1_4"><i><font color="Green" title="E46">A21</font></i></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></i>;<br/>
<a NAME="E15:41_1_4"/><b>then </b>
<font color="Maroon" title="c12">l</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">s</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">l</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>
<a NAME="E16:41_1_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">s</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">l</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">l</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<b>hence </b><a NAME="E17:41_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan6.html#E3:41_1_4"><i><font color="Green" title="E45">A20</font></i></a>, <a class="ref" href="real_2.html#T127" title="REAL_2:th.127">REAL_2:127</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E45">A23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_1_5"/><i><font color="Green" title="E46">A24</font></i>: 
<font color="Maroon" title="c12">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E2:41_1_5"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c11">xp</font> &amp; <font color="Maroon" title="c11">xp</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 )
 <b>by </b><i><a class="txt" href="jordan6.html#E5:41"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="jordan6.html#E7:41_1"><i><font color="Green" title="E37">A12</font></i></a>, <a class="txt" href="jordan6.html#E20:41_1"><i><font color="Green" title="E41">A16</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/><a NAME="E3:41_1_5"/><b>then </b>
<font color="Maroon" title="c11">xp</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">l</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="jordan6.html#E1:41_1_5"><i><font color="Green" title="E46">A24</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/><b>then consider </b><font color="Maroon" title="c15">r5</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="E5:41_1_5"/><i><font color="Green" title="E47">A25</font></i>: 
( <font color="Maroon" title="c15">r5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Q</font> &amp; <font color="Maroon" title="c15">r5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">l</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">xp</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">l</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan6.html#E22:41_1"><i><font color="Green" title="E43">A18</font></i></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">r5</font> = <font color="Maroon" title="c15">r5</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/><a NAME="E7:41_1_5"/><i><font color="Green" title="E48">A26</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">s5</font> where <font color="Olive" title="b1">s5</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Olive" title="b1">s5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c16">r5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">s5</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:41_1_5_1"/>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s5</font> where <font color="Olive" title="b1">s5</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Olive" title="b1">s5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c16">r5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">s5</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:41_1_5_1"/><b>then </b>
 ex <font color="Olive" title="b1">s5</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">s5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y</font> &amp; <font color="Olive" title="b1">s5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c16">r5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">s5</font> )
 
;<br/>
<b>hence </b><a NAME="E3:41_1_5_1"/>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c17">P5</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">s5</font> where <font color="Olive" title="b1">s5</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Olive" title="b1">s5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c16">r5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">s5</font> ) </span>}</span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E9:41_1_5"/><i><font color="Green" title="E49">A27</font></i>: 
<font color="Maroon" title="c11">xp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">P5</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E5:41_1_5"><i><font color="Green" title="E47">A25</font></i></a></i>;<br/><a NAME="E10:41_1_5"/><i><font color="Green" title="E50">A28</font></i>: 
<font color="Maroon" title="c17">P5</font> is <a href="seq_4.html#V2" title="SEQ_4:attr.2">bounded_below</a>
 <b>by </b><i><a class="txt" href="jordan6.html#E21:41_1"><i><font color="Green" title="E42">A17</font></i></a>, <a class="txt" href="jordan6.html#E7:41_1_5"><i><font color="Green" title="E48">A26</font></i></a>, <a class="ref" href="rcomp_1.html#T3" title="RCOMP_1:th.3">RCOMP_1:3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">l5</font> =  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <font color="Maroon" title="c17">P5</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><b>reconsider </b><font color="Maroon" title="c19">m5</font> = <font color="Maroon" title="c18">l5</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a></i>;<br/><div><i><font color="Green" title="E51">A29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_5_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_1_5_2"/><i><font color="Green" title="E52">A30</font></i>: 
<font color="Maroon" title="c18">l5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">r5</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_5_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_1_5_2_1"/>
<font color="Maroon" title="c18">l5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c16">r5</font>
 ;<br/><a NAME="E2:41_1_5_2_1"/><b>then </b>
<font color="Maroon" title="c16">r5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">l5</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/><b>then consider </b><font color="Maroon" title="c20">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="E4:41_1_5_2_1"/><i><font color="Green" title="E53">A31</font></i>: 
( <font color="Maroon" title="c20">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">P5</font> &amp; <font color="Maroon" title="c20">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">l5</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">r5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">l5</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan6.html#E9:41_1_5"><i><font color="Green" title="E49">A27</font></i></a>, <a class="txt" href="jordan6.html#E10:41_1_5"><i><font color="Green" title="E50">A28</font></i></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c21">s6</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E6:41_1_5_2_1"/><i><font color="Green" title="E54">A32</font></i>: 
( <font color="Maroon" title="c21">s6</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">r</font> &amp; <font color="Maroon" title="c21">s6</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c16">r5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c21">s6</font> )
 <b>by </b><i><a class="txt" href="jordan6.html#E4:41_1_5_2_1"><i><font color="Green" title="E53">A31</font></i></a></i>;<br/><b>thus </b><a NAME="E7:41_1_5_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan6.html#E4:41_1_5_2_1"><i><font color="Green" title="E53">A31</font></i></a>, <a class="txt" href="jordan6.html#E6:41_1_5_2_1"><i><font color="Green" title="E54">A32</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E3:41_1_5_2"/><b>then </b>
<font color="Maroon" title="c18">l5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">r5</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E1:41_1_5_2"><i><font color="Green" title="E52">A30</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c20">r7</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="E5:41_1_5_2"/><i><font color="Green" title="E53">A33</font></i>: 
<font color="Maroon" title="c20">r7</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E6:41_1_5_2"/><i><font color="Green" title="E54">A34</font></i>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c19">m5</font>,<font color="Maroon" title="c20">r7</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">B0</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E17:41_1"><i><font color="Green" title="E40">A15</font></i></a>, <a class="txt" href="jordan6.html#E5:41_1_5"><i><font color="Green" title="E47">A25</font></i></a>, <a class="ref" href="topmetr.html#T22" title="TOPMETR:th.22">TOPMETR:22</a>, <a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">TOPMETR:def 7</a></i>;<br/><b>consider </b><font color="Maroon" title="c21">r9</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="E8:41_1_5_2"/><i><font color="Green" title="E55">A35</font></i>: 
( <font color="Maroon" title="c21">r9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">P5</font> &amp; <font color="Maroon" title="c21">r9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">l5</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c20">r7</font> )
 <b>by </b><i><a class="txt" href="jordan6.html#E9:41_1_5"><i><font color="Green" title="E49">A27</font></i></a>, <a class="txt" href="jordan6.html#E10:41_1_5"><i><font color="Green" title="E50">A28</font></i></a>, <a class="txt" href="jordan6.html#E5:41_1_5_2"><i><font color="Green" title="E53">A33</font></i></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c22">r9</font> = <font color="Maroon" title="c21">r9</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>reconsider </b><font color="Maroon" title="c23">e8</font> = <font color="Maroon" title="c22">r9</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a></i>;<br/><a NAME="E11:41_1_5_2"/>
<font color="Maroon" title="c18">l5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">r9</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E10:41_1_5"><i><font color="Green" title="E50">A28</font></i></a>, <a class="txt" href="jordan6.html#E8:41_1_5_2"><i><font color="Green" title="E55">A35</font></i></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></i>;<br/><a NAME="E12:41_1_5_2"/><b>then </b><i><font color="Green" title="E56">A36</font></i>: 
<font color="Maroon" title="c22">r9</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">l5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/><a NAME="E13:41_1_5_2"/>
<font color="Maroon" title="c22">r9</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">l5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c20">r7</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">l5</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E8:41_1_5_2"><i><font color="Green" title="E55">A35</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E14:41_1_5_2"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">r9</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">l5</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c20">r7</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E12:41_1_5_2"><i><font color="Green" title="E56">A36</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><a NAME="E15:41_1_5_2"/><b>then </b>
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c23">e8</font>,<font color="Maroon" title="c19">m5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c20">r7</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D13" title="METRIC_1:def.13">METRIC_1:def 13</a>, <a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a></i>;<br/><a NAME="E16:41_1_5_2"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c23">e8</font>,<font color="Maroon" title="c19">m5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c20">r7</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/><a NAME="E17:41_1_5_2"/><b>then </b>
<font color="Maroon" title="c23">e8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  :  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c19">m5</font>,<font color="Olive" title="b1">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c20">r7</font> </span>}</span> 
 ;<br/><a NAME="E18:41_1_5_2"/><b>then </b>
<font color="Maroon" title="c23">e8</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="c19">m5</font>,<font color="Maroon" title="c20">r7</font>
 <b>by </b><i><a class="ref" href="metric_1.html#T18" title="METRIC_1:th.18">METRIC_1:18</a></i>;<br/><b>hence </b><a NAME="E19:41_1_5_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan6.html#E5:41_1"><i><font color="Green" title="E35">A10</font></i></a>, <a class="txt" href="jordan6.html#E7:41_1_5"><i><font color="Green" title="E48">A26</font></i></a>, <a class="txt" href="jordan6.html#E6:41_1_5_2"><i><font color="Green" title="E54">A34</font></i></a>, <a class="txt" href="jordan6.html#E8:41_1_5_2"><i><font color="Green" title="E55">A35</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><a NAME="E14:41_1_5"/><i><font color="Green" title="E52">A37</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c16">r5</font> &amp; <font color="Maroon" title="c16">r5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 )
 <b>by </b><i><a class="txt" href="jordan6.html#E5:41"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="jordan6.html#E7:41_1"><i><font color="Green" title="E37">A12</font></i></a>, <a class="txt" href="jordan6.html#E5:41_1_5"><i><font color="Green" title="E47">A25</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/><a NAME="E15:41_1_5"/><i><font color="Green" title="E53">A38</font></i>: 
<font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">r5</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="jordan6.html#E13:41_1_5"><i><font color="Green" title="E51">A29</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/><b>consider </b><font color="Maroon" title="c20">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c17">P5</font>;<br/><a NAME="E17:41_1_5"/><i><font color="Green" title="E54">A39</font></i>: 
<font color="Maroon" title="c20">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">P5</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E9:41_1_5"><i><font color="Green" title="E49">A27</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c21">q1</font> = <font color="Maroon" title="c20">q</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><a NAME="E19:41_1_5"/>
<font color="Maroon" title="c21">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E7:41_1_5"><i><font color="Green" title="E48">A26</font></i></a>, <a class="txt" href="jordan6.html#E17:41_1_5"><i><font color="Green" title="E54">A39</font></i></a></i>;<br/><a NAME="E20:41_1_5"/><b>then </b><i><font color="Green" title="E55">A40</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c21">q1</font> &amp; <font color="Maroon" title="c21">q1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 )
 <b>by </b><i><a class="txt" href="jordan6.html#E5:41"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="jordan6.html#E7:41_1"><i><font color="Green" title="E37">A12</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/><a NAME="E21:41_1_5"/>
<font color="Maroon" title="c18">l5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">q1</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E9:41_1_5"><i><font color="Green" title="E49">A27</font></i></a>, <a class="txt" href="jordan6.html#E10:41_1_5"><i><font color="Green" title="E50">A28</font></i></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></i>;<br/><a NAME="E22:41_1_5"/><b>then </b>
<font color="Maroon" title="c18">l5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 <b>by </b><i><a class="txt" href="jordan6.html#E20:41_1_5"><i><font color="Green" title="E55">A40</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E23:41_1_5"/><b>then </b>
<font color="Maroon" title="c18">l5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="jordan6.html#E13:41_1_5"><i><font color="Green" title="E51">A29</font></i></a>, <a class="txt" href="jordan6.html#E14:41_1_5"><i><font color="Green" title="E52">A37</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/><a NAME="E24:41_1_5"/><b>then </b><i><font color="Green" title="E56">A41</font></i>: 
( <font color="Maroon" title="c18">l5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> or <font color="Maroon" title="c18">l5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Q</font> )
 <b>by </b><i><a class="txt" href="jordan6.html#E5:41"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="jordan6.html#E1:41_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="jordan6.html#E7:41_1"><i><font color="Green" title="E37">A12</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_5_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_1_5_3"/>
<font color="Maroon" title="c18">l5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c22">s5</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="E3:41_1_5_3"/><i><font color="Green" title="E57">A42</font></i>: 
<font color="Maroon" title="c22">s5</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E4:41_1_5_3"/><i><font color="Green" title="E58">A43</font></i>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c19">m5</font>,<font color="Maroon" title="c22">s5</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">P</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E17:41_1"><i><font color="Green" title="E40">A15</font></i></a>, <a class="ref" href="topmetr.html#T22" title="TOPMETR:th.22">TOPMETR:22</a>, <a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">TOPMETR:def 7</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c23">s5</font> = <font color="Maroon" title="c22">s5</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.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="c24">s5'</font> =  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c23">s5</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">r5</font></span>)</span>;<br/><a NAME="E6:41_1_5_3"/><i><font color="Green" title="E59">A44</font></i>: 
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c23">s5</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">r5</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="jordan6.html#E15:41_1_5"><i><font color="Green" title="E53">A38</font></i></a>, <a class="txt" href="jordan6.html#E3:41_1_5_3"><i><font color="Green" title="E57">A42</font></i></a>, <a class="ref" href="xxreal_0.html#T21" title="XXREAL_0:th.21">XXREAL_0:21</a></i>;<br/><a NAME="E7:41_1_5_3"/><b>then </b><i><font color="Green" title="E60">A45</font></i>: 
<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c23">s5</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">r5</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="real_2.html#T127" title="REAL_2:th.127">REAL_2:127</a></i>;<br/><a NAME="E8:41_1_5_3"/><i><font color="Green" title="E61">A46</font></i>: 
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c23">s5</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">r5</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">s5</font>
 <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="E9:41_1_5_3"/><i><font color="Green" title="E62">A47</font></i>: 
<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c23">s5</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">r5</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <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> <font color="Maroon" title="c23">s5</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">r5</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan6.html#E6:41_1_5_3"><i><font color="Green" title="E59">A44</font></i></a>, <a class="ref" href="xreal_1.html#T218" title="XREAL_1:th.218">XREAL_1:218</a></i>;<br/><a NAME="E10:41_1_5_3"/>
 <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c23">s5</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">r5</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">r5</font>
 <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="E11:41_1_5_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c23">s5</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">r5</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">r5</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E9:41_1_5_3"><i><font color="Green" title="E62">A47</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E12:41_1_5_3"/><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> <font color="Maroon" title="c23">s5</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">r5</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c16">r5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">r5</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c16">r5</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E13:41_1_5_3"/><b>then </b><i><font color="Green" title="E63">A48</font></i>: 
<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> <font color="Maroon" title="c23">s5</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">r5</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c16">r5</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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> <font color="Maroon" title="c23">s5</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">r5</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</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> <font color="Maroon" title="c23">s5</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">r5</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c25">e1</font> = <font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</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> <font color="Maroon" title="c23">s5</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">r5</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a></i>;<br/><a NAME="E15:41_1_5_3"/>
<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c23">s5</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">r5</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <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> <font color="Maroon" title="c23">s5</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">r5</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan6.html#E6:41_1_5_3"><i><font color="Green" title="E59">A44</font></i></a>, <a class="ref" href="xreal_1.html#T218" title="XREAL_1:th.218">XREAL_1:218</a></i>;<br/><a NAME="E16:41_1_5_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c23">s5</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">r5</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c23">s5</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E8:41_1_5_3"><i><font color="Green" title="E61">A46</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E17:41_1_5_3"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</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> <font color="Maroon" title="c23">s5</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">r5</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c23">s5</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E7:41_1_5_3"><i><font color="Green" title="E60">A45</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><a NAME="E18:41_1_5_3"/><b>then </b>
<a href="metric_1.html#K7" title="METRIC_1:func.7">real_dist</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c18">l5</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</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> <font color="Maroon" title="c23">s5</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">r5</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c23">s5</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D13" title="METRIC_1:def.13">METRIC_1:def 13</a></i>;<br/><a NAME="E19:41_1_5_3"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c19">m5</font>,<font color="Maroon" title="c25">e1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c23">s5</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a>, <a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a></i>;<br/><a NAME="E20:41_1_5_3"/><b>then </b>
<font color="Maroon" title="c25">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  :  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c19">m5</font>,<font color="Olive" title="b1">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c23">s5</font> </span>}</span> 
 ;<br/><a NAME="E21:41_1_5_3"/><b>then </b>
<font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</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> <font color="Maroon" title="c23">s5</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">r5</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <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="c19">m5</font>,<font color="Maroon" title="c23">s5</font>
 <b>by </b><i><a class="ref" href="metric_1.html#T18" title="METRIC_1:th.18">METRIC_1:18</a></i>;<br/><a NAME="E22:41_1_5_3"/><b>then </b><i><font color="Green" title="E64">A49</font></i>: 
<font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</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> <font color="Maroon" title="c23">s5</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">r5</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s7</font> where <font color="Olive" title="b1">s7</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Olive" title="b1">s7</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c16">r5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">s7</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan6.html#E4:41_1_5_3"><i><font color="Green" title="E58">A43</font></i></a>, <a class="txt" href="jordan6.html#E13:41_1_5_3"><i><font color="Green" title="E63">A48</font></i></a></i>;<br/><a NAME="E23:41_1_5_3"/>
<span class="p1">(<span class="default"><a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c23">s5</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">r5</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="jordan6.html#E6:41_1_5_3"><i><font color="Green" title="E59">A44</font></i></a>, <a class="ref" href="real_2.html#T127" title="REAL_2:th.127">REAL_2:127</a></i>;<br/><a NAME="E24:41_1_5_3"/><b>then </b>
<font color="Maroon" title="c18">l5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">l5</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</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> <font color="Maroon" title="c23">s5</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">r5</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E25:41_1_5_3"/><b>then </b>
<font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</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> <font color="Maroon" title="c23">s5</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">r5</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</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> <font color="Maroon" title="c23">s5</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">r5</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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> <font color="Maroon" title="c23">s5</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c16">r5</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>hence </b><a NAME="E26:41_1_5_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan6.html#E10:41_1_5"><i><font color="Green" title="E50">A28</font></i></a>, <a class="txt" href="jordan6.html#E22:41_1_5_3"><i><font color="Green" title="E64">A49</font></i></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c22">s1</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="E27:41_1_5"/><i><font color="Green" title="E57">A50</font></i>: 
<font color="Maroon" title="c22">s1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E28:41_1_5"/><i><font color="Green" title="E58">A51</font></i>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c19">m5</font>,<font color="Maroon" title="c22">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">B0</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E17:41_1"><i><font color="Green" title="E40">A15</font></i></a>, <a class="txt" href="jordan6.html#E24:41_1_5"><i><font color="Green" title="E56">A41</font></i></a>, <a class="ref" href="topmetr.html#T22" title="TOPMETR:th.22">TOPMETR:22</a>, <a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">TOPMETR:def 7</a></i>;<br/><a NAME="E29:41_1_5"/>
<font color="Maroon" title="c22">s1</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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="jordan6.html#E27:41_1_5"><i><font color="Green" title="E57">A50</font></i></a>, <a class="ref" href="real_2.html#T127" title="REAL_2:th.127">REAL_2:127</a></i>;<br/><b>then consider </b><font color="Maroon" title="c23">r2</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="E31:41_1_5"/><i><font color="Green" title="E59">A52</font></i>: 
( <font color="Maroon" title="c23">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">P5</font> &amp; <font color="Maroon" title="c23">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">l5</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">s1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan6.html#E9:41_1_5"><i><font color="Green" title="E49">A27</font></i></a>, <a class="txt" href="jordan6.html#E10:41_1_5"><i><font color="Green" title="E50">A28</font></i></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c24">r2</font> = <font color="Maroon" title="c23">r2</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/><a NAME="E33:41_1_5"/><i><font color="Green" title="E60">A53</font></i>: 
<font color="Maroon" title="c18">l5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">r2</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E10:41_1_5"><i><font color="Green" title="E50">A28</font></i></a>, <a class="txt" href="jordan6.html#E31:41_1_5"><i><font color="Green" title="E59">A52</font></i></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c25">s3</font> = <font color="Maroon" title="c24">r2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">l5</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><b>reconsider </b><font color="Maroon" title="c26">e1</font> = <font color="Maroon" title="c18">l5</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c25">s3</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a></i>;<br/><a NAME="E36:41_1_5"/><i><font color="Green" title="E61">A54</font></i>: 
<font color="Maroon" title="c24">r2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">l5</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="jordan6.html#E33:41_1_5"><i><font color="Green" title="E60">A53</font></i></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/><a NAME="E37:41_1_5"/><i><font color="Green" title="E62">A55</font></i>: 
<font color="Maroon" title="c24">r2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">l5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">s1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">l5</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E31:41_1_5"><i><font color="Green" title="E59">A52</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E38:41_1_5"/>
<font color="Maroon" title="c22">s1</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="c22">s1</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E27:41_1_5"><i><font color="Green" title="E57">A50</font></i></a>, <a class="ref" href="xreal_1.html#T218" title="XREAL_1:th.218">XREAL_1:218</a></i>;<br/><a NAME="E39:41_1_5"/><b>then </b><i><font color="Green" title="E63">A56</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c25">s3</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">l5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c22">s1</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E37:41_1_5"><i><font color="Green" title="E62">A55</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E40:41_1_5"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c25">s3</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">l5</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c25">s3</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">l5</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E36:41_1_5"><i><font color="Green" title="E61">A54</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><a NAME="E41:41_1_5"/><b>then </b>
<a href="metric_1.html#K7" title="METRIC_1:func.7">real_dist</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">l5</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c25">s3</font></span>)</span>,<font color="Maroon" title="c18">l5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c22">s1</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E39:41_1_5"><i><font color="Green" title="E63">A56</font></i></a>, <a class="ref" href="metric_1.html#D13" title="METRIC_1:def.13">METRIC_1:def 13</a></i>;<br/><a NAME="E42:41_1_5"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c26">e1</font>,<font color="Maroon" title="c19">m5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c22">s1</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a>, <a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a></i>;<br/><a NAME="E43:41_1_5"/><b>then </b>
<font color="Maroon" title="c18">l5</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c25">s3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  :  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c19">m5</font>,<font color="Olive" title="b1">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c22">s1</font> </span>}</span> 
 ;<br/><a NAME="E44:41_1_5"/><b>then </b>
( <font color="Maroon" title="c18">l5</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c25">s3</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="c19">m5</font>,<font color="Maroon" title="c22">s1</font> &amp; <font color="Maroon" title="c18">l5</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c25">s3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">P5</font> )
 <b>by </b><i><a class="txt" href="jordan6.html#E31:41_1_5"><i><font color="Green" title="E59">A52</font></i></a>, <a class="ref" href="metric_1.html#T18" title="METRIC_1:th.18">METRIC_1:18</a></i>;<br/><a NAME="E45:41_1_5"/><b>then </b><i><font color="Green" title="E64">A57</font></i>: 
<font color="Maroon" title="c18">l5</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c25">s3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">P5</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Q</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E28:41_1_5"><i><font color="Green" title="E58">A51</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E46:41_1_5"/>
<font color="Maroon" title="c17">P5</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">P</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">Q</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E7:41_1_5"><i><font color="Green" title="E48">A26</font></i></a>, <a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></i>;<br/><b>hence </b><a NAME="E47:41_1_5"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan6.html#E5:41_1"><i><font color="Green" title="E35">A10</font></i></a>, <a class="txt" href="jordan6.html#E45:41_1_5"><i><font color="Green" title="E64">A57</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c15">q</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c7">Q</font>;<br/>
<a NAME="E29:41_1"/><i><font color="Green" title="E46">A58</font></i>: 
<font color="Maroon" title="c15">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Q</font>
 
;<br/>
<b>reconsider </b><font color="Maroon" title="c16">q1</font> = <font color="Maroon" title="c15">q</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E31:41_1"/><i><font color="Green" title="E47">A59</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c16">q1</font> &amp; <font color="Maroon" title="c16">q1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 )
 
<b>by </b><i><a class="txt" href="jordan6.html#E5:41"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="jordan6.html#E7:41_1"><i><font color="Green" title="E37">A12</font></i></a>, <a class="txt" href="jordan6.html#E29:41_1"><i><font color="Green" title="E46">A58</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/>
<a NAME="E32:41_1"/>
<font color="Maroon" title="c12">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">q1</font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E22:41_1"><i><font color="Green" title="E43">A18</font></i></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></i>;<br/>
<a NAME="E33:41_1"/><b>then </b>
<font color="Maroon" title="c12">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 
<b>by </b><i><a class="txt" href="jordan6.html#E31:41_1"><i><font color="Green" title="E47">A59</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E34:41_1"/><b>then </b>
<font color="Maroon" title="c12">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="txt" href="jordan6.html#E27:41_1"><i><font color="Green" title="E45">A23</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/>
<a NAME="E35:41_1"/><b>then </b>
( <font color="Maroon" title="c14">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A0</font> &amp; <font color="Maroon" title="c8">A0</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 
<b>by </b><i><a class="txt" href="jordan6.html#E5:41"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="jordan6.html#E1:41_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="jordan6.html#E3:41_1"><i><font color="Green" title="E33">A8</font></i></a>, <a class="txt" href="jordan6.html#E7:41_1"><i><font color="Green" title="E37">A12</font></i></a>, <a class="txt" href="jordan6.html#E14:41_1"><i><font color="Green" title="E39">A14</font></i></a>, <a class="txt" href="jordan6.html#E26:41_1"><i><font color="Green" title="E44">A19</font></i></a>, <a class="ref" href="tsep_1.html#T17" title="TSEP_1:th.17">TSEP_1:17</a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">s1</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="E37:41_1"/><i><font color="Green" title="E48">A60</font></i>: 
<font color="Maroon" title="c17">s1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E38:41_1"/><i><font color="Green" title="E49">A61</font></i>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c13">m</font>,<font color="Maroon" title="c17">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">A0</font>
 <b>by </b><i><a class="ref" href="topmetr.html#T22" title="TOPMETR:th.22">TOPMETR:22</a>, <a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">TOPMETR:def 7</a></i>;<br/>
<a NAME="E39:41_1"/>
<font color="Maroon" title="c17">s1</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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jordan6.html#E37:41_1"><i><font color="Green" title="E48">A60</font></i></a>, <a class="ref" href="real_2.html#T127" title="REAL_2:th.127">REAL_2:127</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c18">r2</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="E41:41_1"/><i><font color="Green" title="E50">A62</font></i>: 
( <font color="Maroon" title="c18">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Q</font> &amp; <font color="Maroon" title="c18">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">l</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">s1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan6.html#E22:41_1"><i><font color="Green" title="E43">A18</font></i></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">r2</font> = <font color="Maroon" title="c18">r2</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/>
<a NAME="E43:41_1"/><i><font color="Green" title="E51">A63</font></i>: 
<font color="Maroon" title="c12">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">r2</font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E22:41_1"><i><font color="Green" title="E43">A18</font></i></a>, <a class="txt" href="jordan6.html#E41:41_1"><i><font color="Green" title="E50">A62</font></i></a>, <a class="ref" href="seq_4.html#D5" title="SEQ_4:def.5">SEQ_4:def 5</a></i>;<br/>
<b>set </b><font color="Maroon" title="c20">s3</font> = <font color="Maroon" title="c19">r2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">l</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">e1</font> = <font color="Maroon" title="c12">l</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">r2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">l</font></span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E45:41_1"/><i><font color="Green" title="E52">A64</font></i>: 
<font color="Maroon" title="c19">r2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">l</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jordan6.html#E43:41_1"><i><font color="Green" title="E51">A63</font></i></a>, <a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/>
<a NAME="E46:41_1"/><i><font color="Green" title="E53">A65</font></i>: 
<font color="Maroon" title="c19">r2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">s1</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">l</font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E41:41_1"><i><font color="Green" title="E50">A62</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/>
<a NAME="E47:41_1"/>
<font color="Maroon" title="c17">s1</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="c17">s1</font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E37:41_1"><i><font color="Green" title="E48">A60</font></i></a>, <a class="ref" href="xreal_1.html#T218" title="XREAL_1:th.218">XREAL_1:218</a></i>;<br/>
<a NAME="E48:41_1"/><b>then </b><i><font color="Green" title="E54">A66</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">r2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">l</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c17">s1</font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E46:41_1"><i><font color="Green" title="E53">A65</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E49:41_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">r2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">l</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">r2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">l</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">l</font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E45:41_1"><i><font color="Green" title="E52">A64</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E50:41_1"/><b>then </b>
<a href="metric_1.html#K7" title="METRIC_1:func.7">real_dist</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">l</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">r2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">l</font></span>)</span></span>)</span>,<font color="Maroon" title="c12">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c17">s1</font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E48:41_1"><i><font color="Green" title="E54">A66</font></i></a>, <a class="ref" href="metric_1.html#D13" title="METRIC_1:def.13">METRIC_1:def 13</a></i>;<br/>
<a NAME="E51:41_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c21">e1</font>,<font color="Maroon" title="c13">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c17">s1</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a>, <a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E52:41_1"/><b>then </b>
<font color="Maroon" title="c12">l</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">r2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">l</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  :  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c13">m</font>,<font color="Olive" title="b1">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c17">s1</font> </span>}</span> 
 
;<br/>
<a NAME="E53:41_1"/><b>then </b>
( <font color="Maroon" title="c12">l</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">r2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">l</font></span>)</span> <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="c13">m</font>,<font color="Maroon" title="c17">s1</font> &amp; <font color="Maroon" title="c12">l</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">r2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">l</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Q</font> )
 
<b>by </b><i><a class="txt" href="jordan6.html#E41:41_1"><i><font color="Green" title="E50">A62</font></i></a>, <a class="ref" href="metric_1.html#T18" title="METRIC_1:th.18">METRIC_1:18</a></i>;<br/>
<b>hence </b><a NAME="E54:41_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan6.html#E5:41_1"><i><font color="Green" title="E35">A10</font></i></a>, <a class="txt" href="jordan6.html#E38:41_1"><i><font color="Green" title="E49">A61</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>
<a NAME="E7:41"/><b>then </b>
<a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P7</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 
<b>by </b><i><a class="ref" href="connsp_1.html#T12" title="CONNSP_1:th.12">CONNSP_1:12</a></i>;<br/>
<b>hence </b><a NAME="E8:41"/>
( <font color="Maroon" title="c1">P7</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c1">P7</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> )
 <b>by </b><i><a class="txt" href="jordan6.html#E1:41"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="jordan6.html#E2:41"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="jordan6.html#E3:41"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="connsp_1.html#D3" title="CONNSP_1:def.3">CONNSP_1:def 3</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


</div>
