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<div class="add">

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







<b>thus </b><a NAME="E1:53"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="bhsp_2.html#K5" target="_self">cl_Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> implies <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">z</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="bhsp_2.html#K5" target="_self">cl_Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 ;<br/>

<a NAME="E2:53_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Olive">y</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> </span>}</span> 
 
;<br/>
<a NAME="E3:53_1"/><b>then </b>
 ex <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> &amp; <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Olive">y</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> )
 
;<br/>
<b>hence </b><a NAME="E4:53_1"/>
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">z</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 ;<br/>


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

<b>assume </b><a NAME="E2:53"/>
<span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">z</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 ;<br/>

<a NAME="E3:53"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y</font> where <font color="Olive">y</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Olive">y</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E4:53"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="bhsp_2.html#K5" target="_self">cl_Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 ;<br/>


</div>
