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<div class="add">

<b>let </b><font color="Maroon">X</font> be   <a href="csspace.html#NM1">ComplexUnitarySpace</a>;<br/><b>let </b><font color="Maroon">w</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="E1:47"/>
( <font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="clvect_2.html#K4" target="_self">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> implies  <a href="csspace.html#K15">dist</a> <font color="Maroon">x</font>,<font color="Maroon">w</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:47_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="clvect_2.html#K4" target="_self">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 ;<br/>

<a NAME="E2:47_1"/><b>then </b>
<span class="p1"><a href="csspace.html#K13">||.</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">w</font></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:47_1"/>
 <a href="csspace.html#K15">dist</a> <font color="Maroon">x</font>,<font color="Maroon">w</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="csspace.html#D16">CSSPACE:def 16</a></i>;<br/>


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

<b>assume </b><a NAME="E2:47"/>
 <a href="csspace.html#K15">dist</a> <font color="Maroon">x</font>,<font color="Maroon">w</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 ;<br/>

<a NAME="E3:47"/><b>then </b>
<span class="p1"><a href="csspace.html#K13">||.</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">w</font></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="csspace.html#D16">CSSPACE:def 16</a></i>;<br/>
<b>hence </b><a NAME="E4:47"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="clvect_2.html#K4" target="_self">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 ;<br/>


</div>
