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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="csspace.html#NM1" title="CSSPACE:NM.1">ComplexUnitarySpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">w</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b4">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clvect_2.html#K4" title="CLVECT_2:func.4">Ball</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b4">r</font> &amp; <font color="Olive" title="b3">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clvect_2.html#K4" title="CLVECT_2:func.4">Ball</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b4">r</font> holds <br/> <a href="csspace.html#K15" title="CSSPACE:func.15">dist</a> <font color="Olive" title="b1">y</font>,<font color="Olive" title="b3">w</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b4">r</font></span><br/><b>let </b><font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">w</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clvect_2.html#K4" title="CLVECT_2:func.4">Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">r</font> &amp; <font color="Maroon" title="c4">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clvect_2.html#K4" title="CLVECT_2:func.4">Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">r</font> holds <br/> <a href="csspace.html#K15" title="CSSPACE:func.15">dist</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c4">w</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">r</font></span><br/><b>let </b><font color="Maroon" title="c5">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clvect_2.html#K4" title="CLVECT_2:func.4">Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">r</font> &amp; <font color="Maroon" title="c4">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clvect_2.html#K4" title="CLVECT_2:func.4">Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">r</font> implies  <a href="csspace.html#K15" title="CSSPACE:func.15">dist</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c4">w</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c5">r</font> )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:49"/><i><font color="Green" title="E30">A1</font></i>: 
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clvect_2.html#K4" title="CLVECT_2:func.4">Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">r</font>
 <b>and </b><br/><a NAME="E2:49"/><i><font color="Green" title="E31">A2</font></i>: 
<font color="Maroon" title="c4">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clvect_2.html#K4" title="CLVECT_2:func.4">Ball</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="csspace.html#K15" title="CSSPACE:func.15">dist</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c4">w</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c5">r</font></span><br/>

<a NAME="E3:49"/><i><font color="Green" title="E32">A3</font></i>: 
 <a href="csspace.html#K15" title="CSSPACE:func.15">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font>
 
<b>by </b><i><a class="txt" href="#E1:49"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="clvect_2.html#T41" target="_self" title="CLVECT_2:th.41">Th41</a></i>;<br/>
<a NAME="E4:49"/><i><font color="Green" title="E33">A4</font></i>: 
 <a href="csspace.html#K15" title="CSSPACE:func.15">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">w</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font>
 
<b>by </b><i><a class="txt" href="#E2:49"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="clvect_2.html#T41" target="_self" title="CLVECT_2:th.41">Th41</a></i>;<br/>
<a NAME="E5:49"/><i><font color="Green" title="E34">A5</font></i>: 
<span class="p1">(<span class="default"><a href="csspace.html#K15" title="CSSPACE:func.15">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K15" title="CSSPACE:func.15">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K15" title="CSSPACE:func.15">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">w</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:49"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E6:49"/>
<font color="Maroon" title="c5">r</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K15" title="CSSPACE:func.15">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">r</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c5">r</font>
 
<b>by </b><i><a class="txt" href="#E4:49"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E7:49"/><b>then </b><i><font color="Green" title="E35">A6</font></i>: 
<span class="p1">(<span class="default"><a href="csspace.html#K15" title="CSSPACE:func.15">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K15" title="CSSPACE:func.15">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">w</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c5">r</font>
 
<b>by </b><i><a class="txt" href="#E5:49"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:49"/>
 <a href="csspace.html#K15" title="CSSPACE:func.15">dist</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c4">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="csspace.html#K15" title="CSSPACE:func.15">dist</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K15" title="CSSPACE:func.15">dist</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">w</font></span>)</span>
 
<b>by </b><i><a class="ref" href="csspace.html#T54" title="CSSPACE:th.54">CSSPACE:54</a></i>;<br/>
<b>hence </b><a NAME="E9:49"/>
 <a href="csspace.html#K15" title="CSSPACE:func.15">dist</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c4">w</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="txt" href="#E7:49"><i><font color="Green" title="E35">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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