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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="rlvect_1.html#NM1">VECTOR</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>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:52"/><i><font color="Green">E23</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E2:52"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>assume </b><a NAME="E3:52"/>
not  <a href="bhsp_2.html#K6">Sphere</a> <font color="Maroon">v</font>,<font color="Maroon">r</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:52"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bhsp_2.html#K6">Sphere</a> <font color="Maroon">v</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E6:52"/>
 <a href="bhsp_2.html#K6">Sphere</a> <font color="Maroon">v</font>,<font color="Maroon">r</font> <a href="hidden.html#R1">=</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">V</font> : <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</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="hidden.html#R1">=</a> <font color="Maroon">r</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="bhsp_2.html#D7">BHSP_2:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon">w</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E8:52"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> &amp; <span class="p1"><a href="bhsp_1.html#K3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">w</font></span>)</span></span><a href="bhsp_1.html#K3">.||</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> )
 <b>by </b><i/>;<br/>
<a NAME="E9:52"/>
<font color="Maroon">v</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">V</font>
 
<b>by </b><i>, , <a class="ref" href="bhsp_1.html#T32">BHSP_1:32</a></i>;<br/>
<b>hence </b><a NAME="E10:52"/>
contradiction
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


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