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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">x</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="b3">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="b3">r</font>) holds <br/><font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clvect_2.html#K5" title="CLVECT_2:func.5">cl_Ball</a> (<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">r</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">y</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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>) holds <br/><font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clvect_2.html#K5" title="CLVECT_2:func.5">cl_Ball</a> (<font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">r</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c4">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c4">r</font>) implies <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clvect_2.html#K5" title="CLVECT_2:func.5">cl_Ball</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font>) )</span><br/>







<span class="kw">assume </span><a NAME="E1:56"/>
<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="c4">r</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K5" title="CLVECT_2:func.5">cl_Ball</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font>)</span><br/>

<a NAME="E2:56"/><span class="kw">then </span>
<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="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c2">y</font></span>)</span></span><a href="csspace.html#K13" title="CSSPACE:func.13">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="clvect_2.html#T40" target="_self" title="CLVECT_2:th.40">Th40</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:56"/>
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="clvect_2.html#K5" title="CLVECT_2:func.5">cl_Ball</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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