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<b>let </b><font color="Maroon">V</font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">p</font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<b>assume </b><a NAME="E1:38"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E2:38"/>
for <font color="Olive">u</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">V</font>  st <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="bhsp_2.html#K4">Ball</a> <font color="Maroon">v</font>,<font color="Maroon">r</font> holds <br/><font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="bhsp_2.html#K4">Ball</a> <font color="Maroon">v</font>,<font color="Maroon">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">V</font>;<br/>

<b>assume </b><a NAME="E1:38_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="bhsp_2.html#K4">Ball</a> <font color="Maroon">v</font>,<font color="Maroon">r</font>
 ;<br/>

<a NAME="E2:38_1"/><b>then </b>
 <a href="bhsp_1.html#K5">dist</a> <font color="Maroon">v</font>,<font color="Maroon">u</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="bhsp_2.html#T41">BHSP_2:41</a></i>;<br/>
<a NAME="E3:38_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="bhsp_1.html#K5">dist</a> <font color="Maroon">v</font>,<font color="Maroon">u</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="txt" href="rusub_5.html#E1:38"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E4:38_1"/><b>then </b>
 <a href="bhsp_1.html#K5">dist</a> <font color="Maroon">v</font>,<font color="Maroon">u</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<b>hence </b><a NAME="E5:38_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="bhsp_2.html#K4">Ball</a> <font color="Maroon">v</font>,<font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="bhsp_2.html#T41">BHSP_2:41</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:38"/>
 <a href="bhsp_2.html#K4">Ball</a> <font color="Maroon">v</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="bhsp_2.html#K4">Ball</a> <font color="Maroon">v</font>,<font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/>


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