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<b>let </b><font color="Maroon">X</font> be   <a href="csspace.html#NM1">ComplexUnitarySpace</a>;<br/><b>let </b><font color="Maroon">y</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">w</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>assume </b><b>that </b><br/><a NAME="E1:49"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">y</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>
 <b>and </b><br/><a NAME="E2:49"/><i><font color="Green">E44</font></i>: 
<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="E3:49"/><i><font color="Green">E46</font></i>: 
 <a href="csspace.html#K15">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:49"/><i><font color="Green">E47</font></i>: 
 <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>, </i>;<br/>
<a NAME="E5:49"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><a href="csspace.html#K15">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K15">dist</a> <font color="Maroon">x</font>,<font color="Maroon">w</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K15">dist</a> <font color="Maroon">x</font>,<font color="Maroon">w</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E6:49"/>
<font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K15">dist</a> <font color="Maroon">x</font>,<font color="Maroon">w</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E7:49"/><b>then </b><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><a href="csspace.html#K15">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K15">dist</a> <font color="Maroon">x</font>,<font color="Maroon">w</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:49"/>
 <a href="csspace.html#K15">dist</a> <font color="Maroon">y</font>,<font color="Maroon">w</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="csspace.html#K15">dist</a> <font color="Maroon">y</font>,<font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K15">dist</a> <font color="Maroon">x</font>,<font color="Maroon">w</font></span>)</span>
 
<b>by </b><i><a class="ref" href="csspace.html#T54">CSSPACE:54</a></i>;<br/>
<b>hence </b><a NAME="E9:49"/>
 <a href="csspace.html#K15">dist</a> <font color="Maroon">y</font>,<font color="Maroon">w</font> <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="real_1.html#K4">*</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


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