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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon" title="c2">z</font>, <font color="Maroon" title="c3">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</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>thus </b><a NAME="E1:57"/>
( <font color="Maroon" title="c2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bhsp_2.html#K6" title="BHSP_2:func.6">Sphere</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font> implies <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">z</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_1"/>
<font color="Maroon" title="c2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bhsp_2.html#K6" title="BHSP_2:func.6">Sphere</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font>
 ;<br/>

<a NAME="E2:57_1"/><b>then </b>
<font color="Maroon" title="c2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> : <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Olive" title="b1">y</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> </span>}</span> 
 
;<br/>
<a NAME="E3:57_1"/><b>then </b>
 ex <font color="Olive" title="b1">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> &amp; <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Olive" title="b1">y</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> )
 
;<br/>
<b>hence </b><a NAME="E4:57_1"/>
<span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">z</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font>
 ;<br/>


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

<b>assume </b><a NAME="E2:57"/>
<span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Maroon" title="c2">z</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font>
 ;<br/>

<a NAME="E3:57"/><b>then </b>
<font color="Maroon" title="c2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> : <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="rlvect_1.html#K6" title="RLVECT_1:func.6">-</a> <font color="Olive" title="b1">y</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E4:57"/>
<font color="Maroon" title="c2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bhsp_2.html#K6" title="BHSP_2:func.6">Sphere</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font>
 ;<br/>


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