<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="csspace.html#NM1" title="CSSPACE:NM.1">ComplexUnitarySpace</a>;<br/><b>let </b><font color="Maroon" title="c2">w</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="E1:58"/>
( <font color="Maroon" title="c2">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clvect_2.html#K6" title="CLVECT_2:func.6">Sphere</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font> implies  <a href="csspace.html#K15" title="CSSPACE:func.15">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">w</font> <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="58_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:58_1"/>
<font color="Maroon" title="c2">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clvect_2.html#K6" title="CLVECT_2:func.6">Sphere</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font>
 ;<br/>

<a NAME="E2:58_1"/><b>then </b>
<span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</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">w</font></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font>
 
<b>by </b><i><a class="ref" href="clvect_2.html#T51" target="_self" title="CLVECT_2:th.51">Th51</a></i>;<br/>
<b>hence </b><a NAME="E3:58_1"/>
 <a href="csspace.html#K15" title="CSSPACE:func.15">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="ref" href="csspace.html#D16" title="CSSPACE:def.16">CSSPACE:def 16</a></i>;<br/>


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

<b>assume </b><a NAME="E2:58"/>
 <a href="csspace.html#K15" title="CSSPACE:func.15">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font>
 ;<br/>

<a NAME="E3:58"/><b>then </b>
<span class="p1"><a href="csspace.html#K13" title="CSSPACE:func.13">||.</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">w</font></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font>
 
<b>by </b><i><a class="ref" href="csspace.html#D16" title="CSSPACE:def.16">CSSPACE:def 16</a></i>;<br/>
<b>hence </b><a NAME="E4:58"/>
<font color="Maroon" title="c2">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clvect_2.html#K6" title="CLVECT_2:func.6">Sphere</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font>
 ;<br/>


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