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<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon" title="c2">v</font>, <font color="Maroon" title="c3">u</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">V</font>;<br/><b>let </b><font color="Maroon" title="c4">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>





<b>assume </b><a NAME="E1:40"/>
<font color="Maroon" title="c3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">r</font>
 ;<br/>

<a NAME="E2:40"/><b>then </b><i><font color="Green" title="E25">A1</font></i>: 
 <a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">u</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font>
 
<b>by </b><i><a class="ref" href="bhsp_2.html#T41" title="BHSP_2:th.41">BHSP_2:41</a></i>;<br/>
<b>thus </b><a NAME="E3:40"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp;  <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c3">u</font>,<font color="Olive" title="b1">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">r</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c5">p</font> = <font color="Maroon" title="c4">r</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">u</font></span>)</span>;<br/>
<a NAME="E1:40_1"/><i><font color="Green" title="E26">A2</font></i>: 
for <font color="Olive" title="b1">w</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c3">u</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">u</font></span>)</span></span>)</span> holds <br/><font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">w</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">V</font>;<br/>

<b>assume </b><a NAME="E1:40_1_1"/>
<font color="Maroon" title="c6">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c3">u</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">u</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:40_1_1"/><b>then </b>
 <a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c6">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">u</font></span>)</span>
 
<b>by </b><i><a class="ref" href="bhsp_2.html#T41" title="BHSP_2:th.41">BHSP_2:41</a></i>;<br/>
<a NAME="E3:40_1_1"/><b>then </b><i><font color="Green" title="E27">A3</font></i>: 
<span class="p1">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">u</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c6">w</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
<a NAME="E4:40_1_1"/>
<span class="p1">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">u</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c6">w</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c6">w</font>
 
<b>by </b><i><a class="ref" href="bhsp_1.html#T42" title="BHSP_1:th.42">BHSP_1:42</a></i>;<br/>
<a NAME="E5:40_1_1"/><b>then </b>
 <a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c6">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font>
 
<b>by </b><i><a class="txt" href="rusub_5.html#E3:40_1_1"><i><font color="Green" title="E27">A3</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="E6:40_1_1"/>
<font color="Maroon" title="c6">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="ref" href="bhsp_2.html#T41" title="BHSP_2:th.41">BHSP_2:41</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon" title="c4">r</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">u</font></span>)</span>
;<br/>

<b>thus </b><a NAME="E2:40_1"/>
<font color="Maroon" title="c4">r</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">u</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="rusub_5.html#E2:40"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></i>;<br/>

<b>thus </b><a NAME="E3:40_1"/>
 <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c3">u</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">u</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="txt" href="rusub_5.html#E1:40_1"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></i>;<br/>


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


</div>
