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<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:10_1_1_1"/><i><font color="Green" title="E7">A1</font></i>: 
<font color="Maroon" title="c2">V</font> is <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1_1_1_1"/>
( <font color="Maroon" title="c2">V</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:10_1_1_1_1_1"/><i><font color="Green" title="E8">A2</font></i>: 
<font color="Maroon" title="c2">V</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c3">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:10_1_1_1_1_1"/><i><font color="Green" title="E9">A3</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font>
 <b>and </b><br/><a NAME="E4:10_1_1_1_1_1"/><i><font color="Green" title="E10">A4</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> holds <br/> <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="txt" href="metric_6.html#E1:10_1_1_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="tbsp_1.html#D9" title="TBSP_1:def.9">TBSP_1:def 9</a></i>;<br/><b>consider </b><font color="Maroon" title="c4">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E6:10_1_1_1_1_1"/><i><font color="Green" title="E11">A5</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="metric_6.html#E1:10_1_1_1_1_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/><b>take </b><font color="Maroon" title="c5">s</font> = 2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c3">r</font>;<br/><b>take </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c4">x</font>;<br/><b>thus </b><a NAME="E7:10_1_1_1_1_1"/>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">s</font>
 <b>by </b><i><a class="txt" href="metric_6.html#E3:10_1_1_1_1_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="real_2.html#T129" title="REAL_2:th.129">REAL_2:129</a></i>;<br/><b>thus </b><a NAME="E8:10_1_1_1_1_1"/>
<font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c5">s</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:10_1_1_1_1_1_1"/><i><font color="Green" title="E12">A6</font></i>: 
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c8">v</font> = <font color="Maroon" title="c7">u</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E3:10_1_1_1_1_1_1"/><i><font color="Green" title="E13">A7</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c8">v</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="txt" href="metric_6.html#E4:10_1_1_1_1_1"><i><font color="Green" title="E10">A4</font></i></a>, <a class="txt" href="metric_6.html#E6:10_1_1_1_1_1"><i><font color="Green" title="E11">A5</font></i></a>, <a class="txt" href="metric_6.html#E1:10_1_1_1_1_1_1"><i><font color="Green" title="E12">A6</font></i></a></i>;<br/>
<a NAME="E4:10_1_1_1_1_1_1"/>
1 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">s</font>
 
<b>by </b><i><a class="txt" href="metric_6.html#E3:10_1_1_1_1_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/>
<a NAME="E5:10_1_1_1_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c8">v</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">s</font>
 
<b>by </b><i><a class="txt" href="metric_6.html#E3:10_1_1_1_1_1_1"><i><font color="Green" title="E13">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E6:10_1_1_1_1_1_1"/>
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c5">s</font>
 <b>by </b><i><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:10_1_1_1_1_2"/><i><font color="Green" title="E8">A8</font></i>: 
<font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c3">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/><b>take </b><font color="Maroon" title="c4">r</font> = 1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2;<br/><b>take </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c3">x</font>;<br/><b>thus </b><a NAME="E3:10_1_1_1_1_2"/>
( 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> &amp; <font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">r</font> )
 <b>by </b><i><a class="txt" href="metric_6.html#E1:10_1_1_1_1_2"><i><font color="Green" title="E8">A8</font></i></a>, <a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div>

<b>given </b><font color="Maroon" title="c3">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>, <font color="Maroon" title="c4">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><b> such that </b><a NAME="E2:10_1_1"/><i><font color="Green" title="E7">A9</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font>
 <b>and </b><br/><a NAME="E3:10_1_1"/><i><font color="Green" title="E8">A10</font></i>: 
<font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">r</font>
 ;<br/>

<b>take </b>
2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c3">r</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="tbsp_1.html#D9" title="TBSP_1:def.9">TBSP_1:def 9</a><br/>

<b>thus </b><a NAME="E4:10_1_1"/>
2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="metric_6.html#E2:10_1_1"><i><font color="Green" title="E7">A9</font></i></a>, <a class="ref" href="real_2.html#T129" title="REAL_2:th.129">REAL_2:129</a></i>;<br/>

<b>let </b><font color="Maroon" title="c5">z</font>, <font color="Maroon" title="c6">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>assume </b><a NAME="E5:10_1_1"/>
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font>
 ;<br/>

<a NAME="E6:10_1_1"/><b>then </b><i><font color="Green" title="E9">A11</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="txt" href="metric_6.html#E3:10_1_1"><i><font color="Green" title="E8">A10</font></i></a>, <a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<b>assume </b><a NAME="E7:10_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font>
 ;<br/>

<a NAME="E8:10_1_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="txt" href="metric_6.html#E3:10_1_1"><i><font color="Green" title="E8">A10</font></i></a>, <a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<a NAME="E9:10_1_1"/><b>then </b><i><font color="Green" title="E10">A12</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">y</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">r</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="txt" href="metric_6.html#E6:10_1_1"><i><font color="Green" title="E9">A11</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E10:10_1_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c6">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">x</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="metric_1.html#T4" title="METRIC_1:th.4">METRIC_1:4</a></i>;<br/>
<b>hence </b><a NAME="E11:10_1_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c6">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="txt" href="metric_6.html#E9:10_1_1"><i><font color="Green" title="E10">A12</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


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