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<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>let </b><font color="Maroon">x'</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:10"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<b>thus </b><a NAME="E2:10"/>
 <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> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font></span>)</span></span><a href="rcomp_1.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="10_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:10_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">y</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>
 ;<br/>

<b>then reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#K8">RealSpace</a>  ;<br/>
<a NAME="E3:10_1"/><i><font color="Green">E38</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x2</font> = <font color="Maroon">x</font>, <font color="Maroon">y2</font> = <font color="Maroon">y1</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y1</font> = 
<a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font>
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a>, <a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">y2</font></span>)</span>
<b>by </b><i><a class="ref" href="metric_1.html#D13">METRIC_1:def 13</a></i>
<br/>.= 
 <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">y2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x2</font></span>)</span></span>)</span>

<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">y2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x2</font></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
;<br/>
<b>hence </b><a NAME="E6:10_1"/>
<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">x'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x'</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="ref" href="rcomp_1.html#T8">RCOMP_1:8</a></i>;<br/>


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

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

<b>assume </b><a NAME="E3:10"/><i><font color="Green">E39</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">x'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">y2</font> = <font color="Maroon">y</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">y2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x'</font></span>)</span> = 
 <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">y2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">x'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">y2</font></span>)</span>

<br/>.= 
<a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">x'</font>,<font color="Maroon">y2</font>
<b>by </b><i><a class="ref" href="metric_1.html#D13">METRIC_1:def 13</a></i>
;<br/>
<a NAME="E6:10"/><b>then </b><i><font color="Green">E40</font></i>: 
<a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">x'</font>,<font color="Maroon">y2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T8">RCOMP_1:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x'</font>, <font color="Maroon">y1</font> = <font color="Maroon">y2</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E8:10"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">x'</font>,<font color="Maroon">y2</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a>, <a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<b>hence </b><a NAME="E9:10"/>
<font color="Maroon">y</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>
 <b>by </b><i>, , <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>


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