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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>;<br/><b>let </b><font color="Maroon" title="c2">u</font>, <font color="Maroon" title="c3">v</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">V</font>;<br/><b>let </b><font color="Maroon" title="c5">r</font>, <font color="Maroon" title="c6">p</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>





<b>assume </b><a NAME="E1:41"/>
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c5">r</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c6">p</font></span>)</span>
 ;<br/>

<a NAME="E2:41"/><b>then </b><i><font color="Green" title="E25">A1</font></i>: 
( <font color="Maroon" title="c3">v</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">u</font>,<font color="Maroon" title="c5">r</font> &amp; <font color="Maroon" title="c3">v</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="c4">w</font>,<font color="Maroon" title="c6">p</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:41"/><i><font color="Green" title="E26">A2</font></i>: 
( <font color="Maroon" title="c7">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp;  <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c7">s</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">u</font>,<font color="Maroon" title="c5">r</font> )
 <b>by </b><i><a class="ref" href="rusub_5.html#T35" target="_self" title="RUSUB_5:th.35">Th35</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">t</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E6:41"/><i><font color="Green" title="E27">A3</font></i>: 
( <font color="Maroon" title="c8">t</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp;  <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c8">t</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="c4">w</font>,<font color="Maroon" title="c6">p</font> )
 <b>by </b><i><a class="txt" href="rusub_5.html#E2:41"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="rusub_5.html#T35" target="_self" title="RUSUB_5:th.35">Th35</a></i>;<br/>
<b>take </b><font color="Maroon" title="c9">q</font> =  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c8">t</font>;<br/>

<a NAME="E7:41"/>
( <font color="Maroon" title="c9">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">s</font> &amp; <font color="Maroon" title="c9">q</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="rusub_5.html#E4:41"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="rusub_5.html#E6:41"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a>, <a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/>
<a NAME="E8:41"/><b>then </b>
 <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c9">q</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="c3">v</font>,<font color="Maroon" title="c7">s</font>
 
<b>by </b><i><a class="ref" href="rusub_5.html#T33" target="_self" title="RUSUB_5:th.33">Th33</a></i>;<br/>
<b>hence </b><a NAME="E9:41"/>
 <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c9">q</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">u</font>,<font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="txt" href="rusub_5.html#E4:41"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>

<a NAME="E10:41"/>
( <font color="Maroon" title="c9">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">t</font> &amp; <font color="Maroon" title="c9">q</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="rusub_5.html#E4:41"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="rusub_5.html#E6:41"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a>, <a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/>
<a NAME="E11:41"/><b>then </b>
 <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c9">q</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="c3">v</font>,<font color="Maroon" title="c8">t</font>
 
<b>by </b><i><a class="ref" href="rusub_5.html#T33" target="_self" title="RUSUB_5:th.33">Th33</a></i>;<br/>
<b>hence </b><a NAME="E12:41"/>
 <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c9">q</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="c4">w</font>,<font color="Maroon" title="c6">p</font>
 <b>by </b><i><a class="txt" href="rusub_5.html#E6:41"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>


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