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<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</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">v</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b2">r</font>, <font color="Olive" title="b3">p</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">p</font> holds <br/> <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b3">p</font></span><br/><b>let </b><font color="Maroon" title="c2">v</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">V</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>, <font color="Olive" title="b2">p</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">p</font> holds <br/> <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c2">v</font>,<font color="Olive" title="b1">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c2">v</font>,<font color="Olive" title="b2">p</font></span><br/><b>let </b><font color="Maroon" title="c3">r</font>, <font color="Maroon" title="c4">p</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="c3">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p</font> implies  <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">p</font> )</span><br/>





<b>assume </b><a NAME="E1:38"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">p</font></span><br/>

<a NAME="E2:38"/>
for <font color="Olive" title="b1">u</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">r</font> holds <br/><font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">u</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">r</font> implies <font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">p</font> )</span><br/>

<b>assume </b><a NAME="E1:38_1"/>
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">p</font></span><br/>

<a NAME="E2:38_1"/><b>then </b>
 <a href="bhsp_1.html#K4" title="BHSP_1:func.4">dist</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c5">u</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="ref" href="bhsp_2.html#T41" title="BHSP_2:th.41">BHSP_2:41</a></i>;<br/>
<a NAME="E3:38_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="bhsp_1.html#K4" title="BHSP_1:func.4">dist</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c5">u</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c4">p</font>
 
<b>by </b><i><a class="txt" href="#E1:38"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E4:38_1"/><b>then </b>
 <a href="bhsp_1.html#K4" title="BHSP_1:func.4">dist</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c5">u</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>
<b>hence </b><a NAME="E5:38_1"/>
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="ref" href="bhsp_2.html#T41" title="BHSP_2:th.41">BHSP_2:41</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:38"/>
 <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</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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