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

<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>let </b><font color="Maroon">s</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>





<b>set </b><font color="Maroon">M</font> =  <a href="topmetr.html#K2">Closed-Interval-MSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:3"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E2:3"/><i><font color="Green">E73</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font>
 ;<br/>

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>;<br/>

<b>set </b><font color="Maroon">B</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">a</font>;<br/>
<a NAME="E3:3"/><i><font color="Green">E75</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T14">TOPMETR:14</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">p</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="topmetr.html#T12">TOPMETR:12</a></i>;<br/>
<b>set </b><font color="Maroon">B1</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p1</font>,<font color="Maroon">a</font>;<br/>
<a NAME="E5:3"/><i><font color="Green">E78</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">p1</font>,<font color="Maroon">a</font></span>)</span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> <font color="Maroon">r</font>,<font color="Maroon">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T13">TOPMETR:13</a></i>;<br/>
<a NAME="E6:3"/>
( <font color="Maroon">a</font> is   <a href="real_1.html#NM1">Real</a> &amp; <font color="Maroon">p1</font> is   <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="E7:3"/><b>then </b><i><font color="Green">E79</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p1</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i>, <a class="ref" href="frechet.html#T7">FRECHET:7</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:3_1"/>
( ( <font color="Maroon">p1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">p1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> ) or ( <font color="Maroon">p1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">p1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font> ) or ( <font color="Maroon">p1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s</font> &amp; <font color="Maroon">p1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> ) or ( <font color="Maroon">p1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s</font> &amp; <font color="Maroon">p1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:3_1_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">p1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E2:3_1_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">p1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><a NAME="E3:3_1_1"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font></span>)</span></span><a href="rcomp_2.html#K1">.[</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:3_1_1_1"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font></span>)</span></span><a href="rcomp_2.html#K1">.[</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</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="E1:3_1_1_1_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">a</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> ;<br/>
<a NAME="E3:3_1_1_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p1</font>,<font color="Maroon">a</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:3_1_1_1_1"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<a NAME="E5:3_1_1_1_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<b>hence </b><a NAME="E6:3_1_1_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font></span>)</span></span><a href="rcomp_2.html#K1">.[</a></span>
 <b>by </b><i>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">b</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="E2:3_1_1_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font></span>)</span></span><a href="rcomp_2.html#K1">.[</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E4:3_1_1_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/>
<a NAME="E5:3_1_1_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/>
<a NAME="E6:3_1_1_1"/><b>then </b>
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:3_1_1_1"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></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="E8:3_1_1_1"/>
<font color="Maroon">p1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:3_1_1_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p1</font>,<font color="Maroon">a</font>
 
<b>by </b><i>, , <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<b>hence </b><a NAME="E10:3_1_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">a</font>
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:3_1_1"/>
(  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span> or  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font></span>)</span></span><a href="rcomp_2.html#K1">.[</a></span> or  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> or  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:3_1_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">p1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E2:3_1_2"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">p1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><a NAME="E3:3_1_2"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:3_1_2_1"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">b</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="E2:3_1_2_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">b</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">p1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E4:3_1_2_1"/>
<font color="Maroon">p1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<a NAME="E5:3_1_2_1"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:3_1_2_1"/>
<font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<a NAME="E7:3_1_2_1"/><b>then </b>
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:3_1_2_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></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/>
<b>hence </b><a NAME="E9:3_1_2_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">a</font>
 <b>by </b><i>, , , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:3_1_2"/>
(  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span> or  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font></span>)</span></span><a href="rcomp_2.html#K1">.[</a></span> or  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> or  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:3_1_3"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">p1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E2:3_1_3"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">p1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><a NAME="E3:3_1_3"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_3_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:3_1_3_1"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">b</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="E2:3_1_3_1"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E4:3_1_3_1"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E5:3_1_3_1"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E6:3_1_3_1"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font>
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:3_1_3_1"/>
<font color="Maroon">p1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:3_1_3_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p1</font>,<font color="Maroon">a</font>
 
<b>by </b><i>, , <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<b>hence </b><a NAME="E9:3_1_3_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">a</font>
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:3_1_3"/>
(  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span> or  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font></span>)</span></span><a href="rcomp_2.html#K1">.[</a></span> or  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> or  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:3_1_4"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">p1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E2:3_1_4"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">p1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><a NAME="E3:3_1_4"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_4_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:3_1_4_1"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_4_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</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="E1:3_1_4_1_1"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">a</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> ;<br/>
<a NAME="E3:3_1_4_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p1</font>,<font color="Maroon">a</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:3_1_4_1_1"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">p1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<a NAME="E5:3_1_4_1_1"/>
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 
<b>by </b><i>, , <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<b>hence </b><a NAME="E6:3_1_4_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span>
 <b>by </b><i>, <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">b</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="E2:3_1_4_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E4:3_1_4_1"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/>
<a NAME="E5:3_1_4_1"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">p1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/>
<a NAME="E6:3_1_4_1"/><b>then </b>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:3_1_4_1"/><b>then </b><i><font color="Green">E106</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></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="E8:3_1_4_1"/>
<font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font>
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:3_1_4_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p1</font>,<font color="Maroon">a</font>
 
<b>by </b><i>, , <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<b>hence </b><a NAME="E10:3_1_4_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">a</font>
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:3_1_4"/>
(  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K1">.]</a></span> or  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font></span>)</span></span><a href="rcomp_2.html#K1">.[</a></span> or  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">s</font></span><a href="rcomp_2.html#K2">.]</a></span> or  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
