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

<b>let </b><font color="Maroon">S</font> be   <a href="normsp_1.html#NM2">sequence</a> of <a href="topmetr.html#K3">R^1</a> ;<br/>

<b>assume </b><a NAME="E1:12"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">n</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">n</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon">B</font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">S</font> as   <a href="struct_0.html#NM2">Subset</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#D7">TOPMETR:def 7</a></i>;<br/>
<a NAME="E3:12"/>
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="subset_1.html#K3">`</a>  holds <br/> ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">x</font>,<font color="Olive">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="subset_1.html#K3">`</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a> ;<br/>

<b>assume </b><a NAME="E1:12_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_1_1"/>
( <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_1"><b>suppose </b></a><a NAME="E1:12_1_1_1"/>
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:12_1_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">C</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> as   <a href="struct_0.html#NM2">Subset</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/><a NAME="E4:12_1_1_1"/>
<font color="Maroon">C</font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E5:12_1_1_1"/><b>then </b>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#T44">SUBSET_1:44</a></i>;<br/><b>hence </b><a NAME="E6:12_1_1_1"/>
 ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Olive">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="subset_1.html#K3">`</a>  )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_2"><b>suppose </b></a><a NAME="E1:12_1_1_2"/><i><font color="Green">E39</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font>;<br/><a NAME="E3:12_1_1_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="real_1.html#NM1">Real</a> ;<br/><b>consider </b><font color="Maroon">n1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:12_1_1_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">n1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E7:12_1_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n1</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">n1</font> = <font color="Maroon">n1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">r</font> =  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">y</font></span>)</span> as   <a href="real_1.html#NM1">Real</a> ;<br/><a NAME="E10:12_1_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_1_1_2_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:12_1_1_2_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:12_1_1_2_1"/><b>then </b>
<font color="Maroon">B</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#T4">XBOOLE_0:4</a></i>;<br/>
<b>hence </b><a NAME="E4:12_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="subset_1.html#T44">SUBSET_1:44</a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:12_1_1_2"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">x1</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">y</font></span>)</span>
 ;<br/><a NAME="E12:12_1_1_2"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="complex1.html#T133">COMPLEX1:133</a></i>;<br/><a NAME="E13:12_1_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="frechet.html#E1"><i><font color="Green">Lemma17</font></i></a>, <a class="ref" href="complex1.html#T133">COMPLEX1:133</a></i>;<br/><a NAME="E14:12_1_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> 4
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T8">RCOMP_1:8</a></i>;<br/><a NAME="E15:12_1_1_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K6">/</a> 4
 ;<br/><a NAME="E16:12_1_1_2"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K6">/</a> 4
 <b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>;<br/><a NAME="E17:12_1_1_2"/><i><font color="Green">E50</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> <a href="xboole_0.html#R1">misses</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">S</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_1_1_2_2"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">S</font>
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:12_1_1_2_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">S</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>consider </b><font color="Maroon">n2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:12_1_1_2_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">n2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E6:12_1_1_2_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="txt" href="frechet.html#E1:12_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="frechet.html#E1:12_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n2</font> = <font color="Maroon">n2</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_1_1_2_2_1"/>
( <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font> or <font color="Maroon">n1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n2</font> or <font color="Maroon">n1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n2</font> )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:12_1_1_2_2_1_1"/>
<font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font>
 ;<br/><div class="add"><a NAME="E2:12_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, <a class="txt" href="frechet.html#E1:12"><i><font color="Green">E19</font></i></a>, <a class="txt" href="frechet.html#E1:12_1"><i><font color="Green">E22</font></i></a>, , </i>;<br/><font color="Maroon">r</font> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default">0 <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x1</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x1</font></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
;<br/><b>hence </b><a NAME="E4:12_1_1_2_2_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="frechet.html#T2" target="_self">Th2</a>, <a class="ref" href="rcomp_1.html#T8">RCOMP_1:8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:12_1_1_2_2_1_2"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n2</font>
 ;<br/><div class="add"><a NAME="E2:12_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:12_1_1_2_2_1_2"/>
<font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n2</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">n2</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i/>;<br/><a NAME="E4:12_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">n2</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n2</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">a2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:12_1_1_2_2_1_2"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">n2</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n2</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> )
 ;<br/><a NAME="E7:12_1_1_2_2_1_2"/>
<font color="Maroon">z</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n2</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E8:12_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E9:12_1_1_2_2_1_2"/>
<font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i/>;<br/><a NAME="E10:12_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">n1</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n1</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">a1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E12:12_1_1_2_2_1_2"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">n1</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n1</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> )
 ;<br/><a NAME="E13:12_1_1_2_2_1_2"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span>
 <b>by </b><i>, <a class="ref" href="frechet.html#T3" target="_self">Th3</a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E14:12_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span>
 <b>by </b><i><a class="ref" href="frechet.html#T2" target="_self">Th2</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E15:12_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">z</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E16:12_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</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="real_1.html#K1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E17:12_1_1_2_2_1_2"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span>
 <b>by </b><i><a class="txt" href="frechet.html#E2"><i><font color="Green">Lemma20</font></i></a>, <a class="ref" href="pcomps_1.html#T1">PCOMPS_1:1</a></i>;<br/><a NAME="E18:12_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span>
 <b>by </b><i><a class="txt" href="frechet.html#E1:12_1"><i><font color="Green">E22</font></i></a></i>;<br/><a NAME="E19:12_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i/>;<br/><a NAME="E20:12_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">x1</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x1</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">a1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E22:12_1_1_2_2_1_2"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">x1</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x1</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> )
 ;<br/><a NAME="E23:12_1_1_2_2_1_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">z</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">x1</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E24:12_1_1_2_2_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">x1</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x1</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">a1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E26:12_1_1_2_2_1_2"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">x1</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x1</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> )
 ;<br/><a NAME="E27:12_1_1_2_2_1_2"/>
<font color="Maroon">y</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E28:12_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">z</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">y</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E29:12_1_1_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>hence </b><a NAME="E30:12_1_1_2_2_1_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_2_2_1_3"><b>suppose </b></a><a NAME="E1:12_1_1_2_2_1_3"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n2</font>
 ;<br/><div class="add"><a NAME="E2:12_1_1_2_2_1_3"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:12_1_1_2_2_1_3"/>
<font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n2</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">n2</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i/>;<br/><a NAME="E4:12_1_1_2_2_1_3"/><b>then </b>
<font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">n2</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n2</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">a2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:12_1_1_2_2_1_3"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">n2</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n2</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> )
 ;<br/><a NAME="E7:12_1_1_2_2_1_3"/>
<font color="Maroon">z</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E8:12_1_1_2_2_1_3"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">z</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="frechet.html#T2" target="_self">Th2</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E9:12_1_1_2_2_1_3"/>
<font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">n1</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i/>;<br/><a NAME="E10:12_1_1_2_2_1_3"/><b>then </b>
<font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">n1</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n1</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">a1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E12:12_1_1_2_2_1_3"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">n1</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n1</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> )
 ;<br/><a NAME="E13:12_1_1_2_2_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">n1</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">y</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span>
 <b>by </b><i>, <a class="ref" href="frechet.html#T3" target="_self">Th3</a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E14:12_1_1_2_2_1_3"/><b>then </b>
<font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E15:12_1_1_2_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">z</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E16:12_1_1_2_2_1_3"/><b>then </b><i><font color="Green">E85</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</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="real_1.html#K1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</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="real_1.html#K1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E17:12_1_1_2_2_1_3"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span>
 <b>by </b><i><a class="txt" href="frechet.html#E2"><i><font color="Green">Lemma20</font></i></a>, <a class="ref" href="pcomps_1.html#T1">PCOMPS_1:1</a></i>;<br/><a NAME="E18:12_1_1_2_2_1_3"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span>
 <b>by </b><i><a class="txt" href="frechet.html#E1:12_1"><i><font color="Green">E22</font></i></a></i>;<br/><a NAME="E19:12_1_1_2_2_1_3"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i/>;<br/><a NAME="E20:12_1_1_2_2_1_3"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">x1</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x1</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">a1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E22:12_1_1_2_2_1_3"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">x1</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x1</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> )
 ;<br/><a NAME="E23:12_1_1_2_2_1_3"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">z</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x1</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E24:12_1_1_2_2_1_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">x1</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x1</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">a1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E26:12_1_1_2_2_1_3"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">x1</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x1</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> )
 ;<br/><a NAME="E27:12_1_1_2_2_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E28:12_1_1_2_2_1_3"/><b>then </b>
<font color="Maroon">z</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E29:12_1_1_2_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E30:12_1_1_2_2_1_3"/><b>then </b>
<font color="Maroon">z</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <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">1 <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="E31:12_1_1_2_2_1_3"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>reconsider </b><font color="Maroon">C</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> as   <a href="struct_0.html#NM2">Subset</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/><a NAME="E19:12_1_1_2"/>
<font color="Maroon">C</font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i/>;<br/><a NAME="E20:12_1_1_2"/><b>then </b>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#T44">SUBSET_1:44</a></i>;<br/><b>hence </b><a NAME="E21:12_1_1_2"/>
 ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Olive">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="subset_1.html#K3">`</a>  )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:12"/><b>then </b>
<span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">S</font></span>)</span> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 
<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="E5:12"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K3">R^1</a> </span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">S</font></span>)</span> is <a href="pre_topc.html#V3">open</a>
 
;<br/>
<b>hence </b><a NAME="E6:12"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">S</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/>


</div>
