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<b>let </b><font color="Maroon">V</font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">M</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>, <font color="Maroon">N</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:44"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">M</font> <a href="hidden.html#R2">in</a>  <a href="rusub_5.html#K4" target="_self">Family_open_set</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E2:44"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">N</font> <a href="hidden.html#R2">in</a>  <a href="rusub_5.html#K4" target="_self">Family_open_set</a> <font color="Maroon">V</font>
 ;<br/>

<a NAME="E3:44"/>
for <font color="Olive">v</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">V</font>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">N</font> holds <br/> ex <font color="Olive">q</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">q</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="bhsp_2.html#K4">Ball</a> <font color="Olive">v</font>,<font color="Olive">q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">N</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">V</font>;<br/>

<b>assume </b><a NAME="E1:44_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">N</font>
 ;<br/>

<a NAME="E2:44_1"/><b>then </b><i><font color="Green">E26</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">p</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:44_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="bhsp_2.html#K4">Ball</a> <font color="Maroon">v</font>,<font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font> )
 <b>by </b><i><a class="ref" href="rusub_5.html#T2" target="_self">Th2</a>, </i>;<br/>
<b>consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:44_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="bhsp_2.html#K4">Ball</a> <font color="Maroon">v</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font> )
 <b>by </b><i>, , </i>;<br/>
<b>take </b><font color="Maroon">q</font> =  <a href="square_1.html#K1">min</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>;<br/>

<a NAME="E7:44_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">q</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i>, <a class="txt" href="rusub_5.html#E6:44_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a>, <a class="ref" href="xxreal_0.html#T15">XXREAL_0:15</a></i>;<br/>
<b>thus </b><a NAME="E8:44_1"/>
<font color="Maroon">q</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="txt" href="rusub_5.html#E6:44_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xxreal_0.html#T15">XXREAL_0:15</a></i>;<br/>

<a NAME="E9:44_1"/>
 <a href="bhsp_2.html#K4">Ball</a> <font color="Maroon">v</font>,<font color="Maroon">q</font> <a href="tarski.html#R1">c=</a>  <a href="bhsp_2.html#K4">Ball</a> <font color="Maroon">v</font>,<font color="Maroon">p</font>
 
<b>by </b><i><a class="txt" href="rusub_5.html#E7:44_1"><i><font color="Green">E33</font></i></a>, </i>;<br/>
<a NAME="E10:44_1"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="bhsp_2.html#K4">Ball</a> <font color="Maroon">v</font>,<font color="Maroon">q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:44_1"/>
( <font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">q</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i>, <a class="txt" href="rusub_5.html#E6:44_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a>, <a class="ref" href="xxreal_0.html#T15">XXREAL_0:15</a></i>;<br/>
<a NAME="E12:44_1"/><b>then </b>
 <a href="bhsp_2.html#K4">Ball</a> <font color="Maroon">v</font>,<font color="Maroon">q</font> <a href="tarski.html#R1">c=</a>  <a href="bhsp_2.html#K4">Ball</a> <font color="Maroon">v</font>,<font color="Maroon">r</font>
 
<b>by </b><i/>;<br/>
<a NAME="E13:44_1"/><b>then </b>
 <a href="bhsp_2.html#K4">Ball</a> <font color="Maroon">v</font>,<font color="Maroon">q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="txt" href="rusub_5.html#E6:44_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E14:44_1"/>
 <a href="bhsp_2.html#K4">Ball</a> <font color="Maroon">v</font>,<font color="Maroon">q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="txt" href="rusub_5.html#E10:44_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:44"/>
<font color="Maroon">M</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">N</font> <a href="hidden.html#R2">in</a>  <a href="rusub_5.html#K4" target="_self">Family_open_set</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/>


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