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

<b>let </b><font color="Maroon">P7</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="borsuk_1.html#K20">I[01]</a> ;<br/>

<b>assume </b><a NAME="E1:41"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">P7</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a>  <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
 ;<br/>

<a NAME="E2:41"/><i><font color="Green">E47</font></i>: 
1 <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E3:41"/><i><font color="Green">E49</font></i>: 
not 1 <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E4:41"/><b>then </b><i><font color="Green">E51</font></i>: 
1 <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E5:41"/><i><font color="Green">E52</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K20">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">P7</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P7</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E6:41"/><b>then </b><i><font color="Green">E54</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K20">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">P7</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P7</font>
 
;<br/>
<a NAME="E7:41"/>
for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K20">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">P7</font></span>)</span>  st  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K20">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">P7</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> <a href="subset_1.html#K4">\/</a> <font color="Olive">B</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K20">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">P7</font></span>)</span> &amp; <font color="Olive">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K20">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">P7</font></span>)</span> &amp; <font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">B</font> is <a href="pre_topc.html#V3">open</a> holds <br/><font color="Olive">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">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">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K20">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">P7</font></span>)</span>, <font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K20">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">P7</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:41_1"/><i><font color="Green">E55</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K20">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">P7</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E2:41_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K20">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">P7</font></span>)</span> &amp; <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K20">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">P7</font></span>)</span> )
 <b>and </b><br/><a NAME="E3:41_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E4:41_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">B</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>assume </b><a NAME="E5:41_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E6:41_1"/><i><font color="Green">E60</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2">.[</a></span> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i/>;<br/>
<i><font color="Green">E78</font></i>: <font color="Maroon">P7</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2">.[</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
<b>by </b><i>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T11">RCOMP_1:11</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T40">XBOOLE_1:40</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2">.[</a></span>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T83">XBOOLE_1:83</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">D1</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon">P1</font> = <font color="Maroon">P7</font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E10:41_1"/>
 <a href="borsuk_1.html#K20">I[01]</a>  <a href="hidden.html#R1">=</a> <a href="topmetr.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">D1</font>
 
<b>by </b><i><a class="ref" href="topmetr.html#T26">TOPMETR:26</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<a NAME="E11:41_1"/><b>then </b><i><font color="Green">E79</font></i>: 
<a href="borsuk_1.html#K20">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">P7</font> <a href="hidden.html#R1">=</a> <a href="topmetr.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">P1</font>
 
<b>by </b><i>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/>
<a NAME="E12:41_1"/>
<font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">r1</font> where <font color="Olive">r1</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r1</font> &amp; <font color="Olive">r1</font> <a href="xxreal_0.html#R1">&lt;</a> 1 ) </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>;<br/>
<a NAME="E13:41_1"/><b>then </b>
<font color="Maroon">P1</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="jordan2b.html#T32">JORDAN2B:32</a></i>;<br/>
<a NAME="E14:41_1"/><b>then </b><i><font color="Green">E80</font></i>: 
<a href="borsuk_1.html#K20">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">P7</font> is  non <a href="struct_0.html#V3">empty</a> <a href="tsep_1.html#V1">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of  <a href="topmetr.html#K3">R^1</a> 
 
<b>by </b><i>, , , <a class="ref" href="jordan6.html#T4" target="_self">Th4</a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a>, <a class="ref" href="tsep_1.html#T16">TSEP_1:16</a></i>;<br/>
<b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">A</font>, <font color="Maroon">Q</font> = <font color="Maroon">B</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i>, <a class="txt" href="jordan6.html#E1:41"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E16:41_1"/>
the <a href="struct_0.html#U1">carrier</a> of <a href="metric_1.html#K8">RealSpace</a>  <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <a href="metric_1.html#K8">RealSpace</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">A0</font> = <font color="Maroon">P</font>, <font color="Maroon">B0</font> = <font color="Maroon">Q</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/>
<a NAME="E18:41_1"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">A0</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">B0</font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i><a class="ref" href="jordan6.html#T3" target="_self">Th3</a>, , , <a class="ref" href="tsep_1.html#T17">TSEP_1:17</a></i>;<br/>
<b>consider </b><font color="Maroon">xp</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">P</font>;<br/>
<b>reconsider </b><font color="Maroon">xp</font> = <font color="Maroon">xp</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E21:41_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">xp</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
;<br/>
<a NAME="E22:41_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">P</font> is <a href="seq_4.html#V2">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">p</font> = 0;<br/><b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:41_1_2_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:41_1_2_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:41_1_2_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">w</font> &amp; <font color="Olive">w</font> <a href="xxreal_0.html#R1">&lt;</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>;<br/><a NAME="E4:41_1_2_1"/><b>then </b>
 ex <font color="Olive">u</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> <a href="xxreal_0.html#R1">&lt;</a> 1 )
 ;<br/><b>hence </b><a NAME="E5:41_1_2_1"/>
<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:41_1_2"/>
<font color="Maroon">P</font> is <a href="seq_4.html#V2">bounded_below</a>
 <b>by </b><i><a class="ref" href="seq_4.html#D2">SEQ_4:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:41_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">Q</font> is <a href="seq_4.html#V2">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">p</font> = 0;<br/><b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:41_1_3_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 ;<br/><a NAME="E2:41_1_3_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:41_1_3_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">w</font> &amp; <font color="Olive">w</font> <a href="xxreal_0.html#R1">&lt;</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>;<br/><a NAME="E4:41_1_3_1"/><b>then </b>
 ex <font color="Olive">u</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> <a href="xxreal_0.html#R1">&lt;</a> 1 )
 ;<br/><b>hence </b><a NAME="E5:41_1_3_1"/>
<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:41_1_3"/>
<font color="Maroon">Q</font> is <a href="seq_4.html#V2">bounded_below</a>
 <b>by </b><i><a class="ref" href="seq_4.html#D2">SEQ_4:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">l</font> =  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">l</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">m</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/>
<a NAME="E27:41_1"/><i><font color="Green">E105</font></i>: 
not <font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">l</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 ;<br/>

<b>then consider </b><font color="Maroon">s</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:41_1_4"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E4:41_1_4"/><i><font color="Green">E119</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">m</font>,<font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B0</font>
 <b>by </b><i>, <a class="ref" href="topmetr.html#T22">TOPMETR:22</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">e1</font> = <font color="Maroon">l</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K6">/</a> 2</span>)</span> as   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E7:41_1_4"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">s</font> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<a NAME="E8:41_1_4"/>
<font color="Maroon">s</font> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T218">XREAL_1:218</a></i>;<br/>
<a NAME="E9:41_1_4"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E5:41"><i><font color="Green">E52</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E10:41_1_4"/><b>then </b>
the <a href="metric_1.html#U1">distance</a> of <a href="metric_1.html#K8">RealSpace</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">m</font>,<font color="Maroon">e1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D13">METRIC_1:def 13</a>, <a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E11:41_1_4"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">m</font>,<font color="Maroon">e1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/>
<a NAME="E12:41_1_4"/><b>then </b>
<font color="Maroon">e1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#K8">RealSpace</a>  :  <a href="metric_1.html#K4">dist</a> <font color="Maroon">m</font>,<font color="Olive">z</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> </span>}</span> 
 
;<br/>
<a NAME="E13:41_1_4"/><b>then </b>
<font color="Maroon">l</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">m</font>,<font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T18">METRIC_1:18</a></i>;<br/>
<a NAME="E14:41_1_4"/><b>then </b>
<font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i>, <a class="txt" href="jordan6.html#E4:41"><i><font color="Green">E51</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<a NAME="E15:41_1_4"/><b>then </b>
<font color="Maroon">l</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E16:41_1_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<b>hence </b><a NAME="E17:41_1_4"/>
contradiction
 <b>by </b><i>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E121</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">E122</font></i>: 
<font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><a NAME="E2:41_1_5"/>
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">xp</font> &amp; <font color="Maroon">xp</font> <a href="xxreal_0.html#R1">&lt;</a> 1 )
 <b>by </b><i>, , , </i>;<br/><a NAME="E3:41_1_5"/><b>then </b>
<font color="Maroon">xp</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="jordan6.html#E4:41"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><b>then consider </b><font color="Maroon">r5</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E5:41_1_5"/><i><font color="Green">E147</font></i>: 
( <font color="Maroon">r5</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">r5</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">xp</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">r5</font> = <font color="Maroon">r5</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E7:41_1_5"/><i><font color="Green">E148</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">s5</font> where <font color="Olive">s5</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Olive">s5</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">r5</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s5</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">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">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

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

<a NAME="E2:41_1_5_1"/><b>then </b>
 ex <font color="Olive">s5</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">s5</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Olive">s5</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">r5</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s5</font> )
 
;<br/>
<b>hence </b><a NAME="E3:41_1_5_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">P5</font> = <span class="p1">{<span class="default"> <font color="Olive">s5</font> where <font color="Olive">s5</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Olive">s5</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">r5</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s5</font> ) </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E9:41_1_5"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">xp</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P5</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E5:41"><i><font color="Green">E52</font></i></a></i>;<br/><a NAME="E10:41_1_5"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">P5</font> is <a href="seq_4.html#V2">bounded_below</a>
 <b>by </b><i><a class="txt" href="jordan6.html#E2:41"><i><font color="Green">E47</font></i></a>, <a class="ref" href="jordan6.html#T5" target="_self">Th5</a>, <a class="ref" href="rcomp_1.html#T3">RCOMP_1:3</a></i>;<br/><b>reconsider </b><font color="Maroon">l5</font> =  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P5</font> as   <a href="real_1.html#NM1">Real</a> ;<br/><b>reconsider </b><font color="Maroon">m5</font> = <font color="Maroon">l5</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/><div><i><font color="Green">E155</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">E156</font></i>: 
<font color="Maroon">l5</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">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">l5</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r5</font>
 ;<br/><a NAME="E2:41_1_5_2_1"/><b>then </b>
<font color="Maroon">r5</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l5</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><b>then consider </b><font color="Maroon">r</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E4:41_1_5_2_1"/><i><font color="Green">E157</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P5</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l5</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">r5</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l5</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan6.html#E6:41"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan6.html#E1:41_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/><b>consider </b><font color="Maroon">s6</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:41_1_5_2_1"/><i><font color="Green">E159</font></i>: 
( <font color="Maroon">s6</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">s6</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">r5</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s6</font> )
 <b>by </b><i><a class="txt" href="jordan6.html#E4:41_1"><i><font color="Green">E58</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"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan6.html#E5:41_1"><i><font color="Green">E59</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E3:41_1_5_2"/><b>then </b>
<font color="Maroon">l5</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r5</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E3:41_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>then consider </b><font color="Maroon">r7</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E5:41_1_5_2"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">r7</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E6:41_1_5_2"/><i><font color="Green">E162</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">m5</font>,<font color="Maroon">r7</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B0</font>
 <b>by </b><i>, <a class="txt" href="jordan6.html#E5:41"><i><font color="Green">E52</font></i></a>, <a class="ref" href="topmetr.html#T22">TOPMETR:22</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/><b>consider </b><font color="Maroon">r9</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E8:41_1_5_2"/><i><font color="Green">E164</font></i>: 
( <font color="Maroon">r9</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P5</font> &amp; <font color="Maroon">r9</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l5</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r7</font> )
 <b>by </b><i><a class="txt" href="jordan6.html#E6:41"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan6.html#E1:41_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan6.html#E4:41_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">r9</font> = <font color="Maroon">r9</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">e8</font> = <font color="Maroon">r9</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/><a NAME="E11:41_1_5_2"/>
<font color="Maroon">l5</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r9</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E1:41_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan6.html#E6:41_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/><a NAME="E12:41_1_5_2"/><b>then </b><i><font color="Green">E166</font></i>: 
<font color="Maroon">r9</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l5</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E13:41_1_5_2"/>
<font color="Maroon">r9</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l5</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">l5</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r7</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">l5</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E6:41_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E14:41_1_5_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">r9</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l5</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r7</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T6" target="_self">Th6</a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E15:41_1_5_2"/><b>then </b>
the <a href="metric_1.html#U1">distance</a> of <a href="metric_1.html#K8">RealSpace</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">e8</font>,<font color="Maroon">m5</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r7</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D13">METRIC_1:def 13</a>, <a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/><a NAME="E16:41_1_5_2"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">e8</font>,<font color="Maroon">m5</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r7</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/><a NAME="E17:41_1_5_2"/><b>then </b>
<font color="Maroon">e8</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#K8">RealSpace</a>  :  <a href="metric_1.html#K4">dist</a> <font color="Maroon">m5</font>,<font color="Olive">z</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r7</font> </span>}</span> 
 ;<br/><a NAME="E18:41_1_5_2"/><b>then </b>
<font color="Maroon">e8</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">m5</font>,<font color="Maroon">r7</font>
 <b>by </b><i><a class="ref" href="metric_1.html#T18">METRIC_1:18</a></i>;<br/><b>hence </b><a NAME="E19:41_1_5_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jordan6.html#T5" target="_self">Th5</a>, <a class="txt" href="jordan6.html#E5:41_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan6.html#E6:41_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><a NAME="E14:41_1_5"/>
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r5</font> &amp; <font color="Maroon">r5</font> <a href="xxreal_0.html#R1">&lt;</a> 1 )
 <b>by </b><i>, , <a class="txt" href="jordan6.html#E5:41"><i><font color="Green">E52</font></i></a>, </i>;<br/><a NAME="E15:41_1_5"/><b>then </b><i><font color="Green">E167</font></i>: 
<font color="Maroon">l5</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="jordan6.html#E2:41_1"><i><font color="Green">E56</font></i></a></i>;<br/><a NAME="E16:41_1_5"/><i><font color="Green">E168</font></i>: 
<font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r5</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="jordan6.html#E2:41_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><b>consider </b><font color="Maroon">q</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">P5</font>;<br/><a NAME="E18:41_1_5"/><i><font color="Green">E169</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P5</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E6:41"><i><font color="Green">E54</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">q1</font> = <font color="Maroon">q</font> as   <a href="real_1.html#NM1">Real</a> ;<br/><a NAME="E20:41_1_5"/>
<font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T5" target="_self">Th5</a>, <a class="txt" href="jordan6.html#E5:41_1"><i><font color="Green">E59</font></i></a></i>;<br/><a NAME="E21:41_1_5"/><b>then </b><i><font color="Green">E170</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">q1</font> <a href="xxreal_0.html#R1">&lt;</a> 1 )
 <b>by </b><i>, , </i>;<br/><a NAME="E22:41_1_5"/>
<font color="Maroon">l5</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q1</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E6:41"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan6.html#E1:41_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/><a NAME="E23:41_1_5"/><b>then </b>
<font color="Maroon">l5</font> <a href="xxreal_0.html#R1">&lt;</a> 1
 <b>by </b><i><a class="txt" href="jordan6.html#E6:41_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E24:41_1_5"/><b>then </b>
<font color="Maroon">l5</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="jordan6.html#E3:41_1"><i><font color="Green">E57</font></i></a>, </i>;<br/><a NAME="E25:41_1_5"/><b>then </b><i><font color="Green">E171</font></i>: 
( <font color="Maroon">l5</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> or <font color="Maroon">l5</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 <b>by </b><i>, <a class="ref" href="jordan6.html#T2" target="_self">Th2</a>, , <a class="ref" href="xboole_0.html#D2">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">l5</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/><b>then consider </b><font color="Maroon">s5</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:41_1_5_3"/><i><font color="Green">E172</font></i>: 
<font color="Maroon">s5</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E4:41_1_5_3"/><i><font color="Green">E173</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">m5</font>,<font color="Maroon">s5</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="topmetr.html#T22">TOPMETR:22</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/><b>reconsider </b><font color="Maroon">s5</font> = <font color="Maroon">s5</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><b>set </b><font color="Maroon">s5'</font> =  <a href="square_1.html#K1">min</a> <font color="Maroon">s5</font>,<span class="p1">(<span class="default"><font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r5</font></span>)</span>;<br/><a NAME="E6:41_1_5_3"/><i><font color="Green">E175</font></i>: 
 <a href="square_1.html#K1">min</a> <font color="Maroon">s5</font>,<span class="p1">(<span class="default"><font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r5</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="jordan6.html#E4:41_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="jordan6.html#T7" target="_self">Th7</a>, <a class="ref" href="xxreal_0.html#T21">XXREAL_0:21</a></i>;<br/><a NAME="E7:41_1_5_3"/><b>then </b><i><font color="Green">E176</font></i>: 
<span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s5</font>,<span class="p2">(<span class="default"><font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r5</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><a NAME="E8:41_1_5_3"/><i><font color="Green">E177</font></i>: 
 <a href="square_1.html#K1">min</a> <font color="Maroon">s5</font>,<span class="p1">(<span class="default"><font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r5</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s5</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/><a NAME="E9:41_1_5_3"/><i><font color="Green">E178</font></i>: 
<span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s5</font>,<span class="p2">(<span class="default"><font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r5</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a>  <a href="square_1.html#K1">min</a> <font color="Maroon">s5</font>,<span class="p1">(<span class="default"><font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r5</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T218">XREAL_1:218</a></i>;<br/><a NAME="E10:41_1_5_3"/>
 <a href="square_1.html#K1">min</a> <font color="Maroon">s5</font>,<span class="p1">(<span class="default"><font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r5</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r5</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T17">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">min</a> <font color="Maroon">s5</font>,<span class="p2">(<span class="default"><font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r5</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r5</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T10" target="_self">Th10</a>, <a class="ref" href="xxreal_0.html#T2">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">min</a> <font color="Maroon">s5</font>,<span class="p3">(<span class="default"><font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r5</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">r5</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r5</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">r5</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E13:41_1_5_3"/><b>then </b><i><font color="Green">E179</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">min</a> <font color="Maroon">s5</font>,<span class="p4">(<span class="default"><font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r5</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">r5</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s5</font>,<span class="p3">(<span class="default"><font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r5</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s5</font>,<span class="p3">(<span class="default"><font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r5</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>reconsider </b><font color="Maroon">e1</font> = <font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s5</font>,<span class="p3">(<span class="default"><font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r5</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> as   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">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">min</a> <font color="Maroon">s5</font>,<span class="p2">(<span class="default"><font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r5</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a>  <a href="square_1.html#K1">min</a> <font color="Maroon">s5</font>,<span class="p1">(<span class="default"><font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r5</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T218">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">min</a> <font color="Maroon">s5</font>,<span class="p2">(<span class="default"><font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r5</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s5</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T9" target="_self">Th9</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E17:41_1_5_3"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s5</font>,<span class="p5">(<span class="default"><font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r5</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s5</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T8" target="_self">Th8</a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E18:41_1_5_3"/><b>then </b>
<a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">l5</font>,<span class="p1">(<span class="default"><font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s5</font>,<span class="p4">(<span class="default"><font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r5</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s5</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D13">METRIC_1:def 13</a></i>;<br/><a NAME="E19:41_1_5_3"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">m5</font>,<font color="Maroon">e1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s5</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a>, <a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/><a NAME="E20:41_1_5_3"/><b>then </b>
<font color="Maroon">e1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#K8">RealSpace</a>  :  <a href="metric_1.html#K4">dist</a> <font color="Maroon">m5</font>,<font color="Olive">z</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s5</font> </span>}</span> 
 ;<br/><a NAME="E21:41_1_5_3"/><b>then </b>
<font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s5</font>,<span class="p3">(<span class="default"><font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r5</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">m5</font>,<font color="Maroon">s5</font>
 <b>by </b><i><a class="ref" href="metric_1.html#T18">METRIC_1:18</a></i>;<br/><a NAME="E22:41_1_5_3"/><b>then </b><i><font color="Green">E180</font></i>: 
<font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s5</font>,<span class="p3">(<span class="default"><font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r5</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">s7</font> where <font color="Olive">s7</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Olive">s7</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">r5</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s7</font> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="jordan6.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E23:41_1_5_3"/>
<span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s5</font>,<span class="p2">(<span class="default"><font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r5</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><a NAME="E24:41_1_5_3"/><b>then </b>
<font color="Maroon">l5</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l5</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s5</font>,<span class="p3">(<span class="default"><font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r5</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E25:41_1_5_3"/><b>then </b>
<font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s5</font>,<span class="p3">(<span class="default"><font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r5</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">l5</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s5</font>,<span class="p4">(<span class="default"><font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r5</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s5</font>,<span class="p3">(<span class="default"><font color="Maroon">l5</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r5</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">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#E1:41_1"><i><font color="Green">E55</font></i></a>, , <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">s1</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E28:41_1_5"/><i><font color="Green">E182</font></i>: 
<font color="Maroon">s1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E29:41_1_5"/><i><font color="Green">E183</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">m5</font>,<font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B0</font>
 <b>by </b><i>, <a class="ref" href="jordan6.html#T6" target="_self">Th6</a>, <a class="ref" href="topmetr.html#T22">TOPMETR:22</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/><a NAME="E30:41_1_5"/>
<font color="Maroon">s1</font> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="jordan6.html#T7" target="_self">Th7</a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><b>then consider </b><font color="Maroon">r2</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E32:41_1_5"/><i><font color="Green">E184</font></i>: 
( <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P5</font> &amp; <font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l5</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan6.html#E6:41"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan6.html#E1:41_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">r2</font> = <font color="Maroon">r2</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E34:41_1_5"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">l5</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E1:41_1"><i><font color="Green">E55</font></i></a>, , <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">s3</font> = <font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l5</font> as   <a href="real_1.html#NM1">Real</a> ;<br/><b>reconsider </b><font color="Maroon">e1</font> = <font color="Maroon">l5</font> <a href="real_1.html#K3">+</a> <font color="Maroon">s3</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/><a NAME="E37:41_1_5"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l5</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="jordan6.html#T8" target="_self">Th8</a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E38:41_1_5"/><i><font color="Green">E188</font></i>: 
<font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l5</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">l5</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">l5</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E39:41_1_5"/>
<font color="Maroon">s1</font> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T7" target="_self">Th7</a>, <a class="ref" href="xreal_1.html#T218">XREAL_1:218</a></i>;<br/><a NAME="E40:41_1_5"/><b>then </b><i><font color="Green">E189</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">l5</font> <a href="real_1.html#K3">+</a> <font color="Maroon">s3</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">l5</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T10" target="_self">Th10</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E41:41_1_5"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l5</font> <a href="real_1.html#K3">+</a> <font color="Maroon">s3</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">l5</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">l5</font> <a href="real_1.html#K3">+</a> <font color="Maroon">s3</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">l5</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T9" target="_self">Th9</a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E42:41_1_5"/><b>then </b>
<a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">l5</font> <a href="real_1.html#K3">+</a> <font color="Maroon">s3</font></span>)</span>,<font color="Maroon">l5</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T11" target="_self">Th11</a>, <a class="ref" href="metric_1.html#D13">METRIC_1:def 13</a></i>;<br/><a NAME="E43:41_1_5"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">e1</font>,<font color="Maroon">m5</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a>, <a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/><a NAME="E44:41_1_5"/><b>then </b>
<font color="Maroon">l5</font> <a href="real_1.html#K3">+</a> <font color="Maroon">s3</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#K8">RealSpace</a>  :  <a href="metric_1.html#K4">dist</a> <font color="Maroon">m5</font>,<font color="Olive">z</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font> </span>}</span> 
 ;<br/><a NAME="E45:41_1_5"/><b>then </b>
( <font color="Maroon">l5</font> <a href="real_1.html#K3">+</a> <font color="Maroon">s3</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">m5</font>,<font color="Maroon">s1</font> &amp; <font color="Maroon">l5</font> <a href="real_1.html#K3">+</a> <font color="Maroon">s3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P5</font> )
 <b>by </b><i>, <a class="ref" href="metric_1.html#T18">METRIC_1:18</a></i>;<br/><a NAME="E46:41_1_5"/><b>then </b><i><font color="Green">E190</font></i>: 
<font color="Maroon">l5</font> <a href="real_1.html#K3">+</a> <font color="Maroon">s3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P5</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E47:41_1_5"/>
<font color="Maroon">P5</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><b>hence </b><a NAME="E48:41_1_5"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">q</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Q</font>;<br/>
<a NAME="E30:41_1"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 
;<br/>
<b>reconsider </b><font color="Maroon">q1</font> = <font color="Maroon">q</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E32:41_1"/><i><font color="Green">E192</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">q1</font> <a href="xxreal_0.html#R1">&lt;</a> 1 )
 
<b>by </b><i>, , <a class="txt" href="jordan6.html#E4:41"><i><font color="Green">E51</font></i></a>, </i>;<br/>
<a NAME="E33:41_1"/>
<font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q1</font>
 
<b>by </b><i>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<a NAME="E34:41_1"/><b>then </b>
<font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;</a> 1
 
<b>by </b><i><a class="txt" href="jordan6.html#E5:41"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E35:41_1"/><b>then </b>
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default">0,1</span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E36:41_1"/><b>then </b>
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A0</font> &amp; <font color="Maroon">A0</font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i>, <a class="ref" href="jordan6.html#T2" target="_self">Th2</a>, <a class="ref" href="jordan6.html#T3" target="_self">Th3</a>, , , <a class="txt" href="jordan6.html#E3:41"><i><font color="Green">E49</font></i></a>, <a class="ref" href="tsep_1.html#T17">TSEP_1:17</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">s1</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E38:41_1"/><i><font color="Green">E193</font></i>: 
<font color="Maroon">s1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E39:41_1"/><i><font color="Green">E194</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">m</font>,<font color="Maroon">s1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A0</font>
 <b>by </b><i><a class="ref" href="topmetr.html#T22">TOPMETR:22</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/>
<a NAME="E40:41_1"/>
<font color="Maroon">s1</font> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="jordan6.html#T5" target="_self">Th5</a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<b>then consider </b><font color="Maroon">r2</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E42:41_1"/><i><font color="Green">E195</font></i>: 
( <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span> )
 <b>by </b><i>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r2</font> = <font color="Maroon">r2</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E44:41_1"/><i><font color="Green">E196</font></i>: 
<font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font>
 
<b>by </b><i>, <a class="txt" href="jordan6.html#E1:41_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<b>set </b><font color="Maroon">s3</font> = <font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l</font>;<br/>
<b>reconsider </b><font color="Maroon">e1</font> = <font color="Maroon">l</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> as   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E46:41_1"/><i><font color="Green">E197</font></i>: 
<font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="jordan6.html#E2:41_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E47:41_1"/><i><font color="Green">E198</font></i>: 
<font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="xcmplx_0.html#K7">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">l</font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E1:41_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E48:41_1"/>
<font color="Maroon">s1</font> <a href="xcmplx_0.html#K7">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font>
 
<b>by </b><i><a class="ref" href="jordan6.html#T5" target="_self">Th5</a>, <a class="ref" href="xreal_1.html#T218">XREAL_1:218</a></i>;<br/>
<a NAME="E49:41_1"/><b>then </b><i><font color="Green">E199</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E4:41_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E50:41_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">l</font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E3:41_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E51:41_1"/><b>then </b>
<a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span></span>)</span>,<font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E5:41_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="metric_1.html#D13">METRIC_1:def 13</a></i>;<br/>
<a NAME="E52:41_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">e1</font>,<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a>, <a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E53:41_1"/><b>then </b>
<font color="Maroon">l</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#K8">RealSpace</a>  :  <a href="metric_1.html#K4">dist</a> <font color="Maroon">m</font>,<font color="Olive">z</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font> </span>}</span> 
 
;<br/>
<a NAME="E54:41_1"/><b>then </b>
( <font color="Maroon">l</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">m</font>,<font color="Maroon">s1</font> &amp; <font color="Maroon">l</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">l</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 
<b>by </b><i><a class="txt" href="jordan6.html#E1:41_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="metric_1.html#T18">METRIC_1:18</a></i>;<br/>
<b>hence </b><a NAME="E55:41_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="jordan6.html#E6:41"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:41"/><b>then </b>
<a href="borsuk_1.html#K20">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">P7</font> is <a href="connsp_1.html#V1">connected</a>
 
<b>by </b><i><a class="ref" href="connsp_1.html#T12">CONNSP_1:12</a></i>;<br/>
<b>hence </b><a NAME="E9:41"/>
( <font color="Maroon">P7</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">P7</font> is <a href="connsp_1.html#V2">connected</a> )
 <b>by </b><i>, , , <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="connsp_1.html#D3">CONNSP_1:def 3</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div>
