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<div class="add">

<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">M</font>, <font color="Maroon" title="c3">N</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:44"/><i><font color="Green" title="E28">A1</font></i>: 
<font color="Maroon" title="c2">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_5.html#K4" title="RUSUB_5:func.4">Family_open_set</a> <font color="Maroon" title="c1">V</font>
 <b>and </b><br/><a NAME="E2:44"/><i><font color="Green" title="E29">A2</font></i>: 
<font color="Maroon" title="c3">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_5.html#K4" title="RUSUB_5:func.4">Family_open_set</a> <font color="Maroon" title="c1">V</font>
 ;<br/>

<a NAME="E3:44"/>
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>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">N</font> holds <br/> ex <font color="Olive" title="b2">q</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">q</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="Olive" title="b1">v</font>,<font color="Olive" title="b2">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">M</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">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" title="c4">v</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>assume </b><a NAME="E1:44_1"/>
<font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">N</font>
 ;<br/>

<a NAME="E2:44_1"/><b>then </b><i><font color="Green" title="E30">A3</font></i>: 
( <font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</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="c5">p</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:44_1"/><i><font color="Green" title="E31">A4</font></i>: 
( <font color="Maroon" title="c5">p</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="c4">v</font>,<font color="Maroon" title="c5">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">M</font> )
 <b>by </b><i><a class="txt" href="rusub_5.html#E1:44"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="rusub_5.html#D5" target="_self" title="RUSUB_5:def.5">Def5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E6:44_1"/><i><font color="Green" title="E32">A5</font></i>: 
( <font color="Maroon" title="c6">r</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="c4">v</font>,<font color="Maroon" title="c6">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">N</font> )
 <b>by </b><i><a class="txt" href="rusub_5.html#E2:44"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="rusub_5.html#E2:44_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="rusub_5.html#D5" target="_self" title="RUSUB_5:def.5">Def5</a></i>;<br/>
<b>take </b><font color="Maroon" title="c7">q</font> =  <a href="square_1.html#K1" title="SQUARE_1:func.1">min</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">r</font>;<br/>

<a NAME="E7:44_1"/><i><font color="Green" title="E33">A6</font></i>: 
( <font color="Maroon" title="c7">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">p</font> &amp; <font color="Maroon" title="c7">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#E4:44_1"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="rusub_5.html#E6:44_1"><i><font color="Green" title="E32">A5</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/>
<b>thus </b><a NAME="E8:44_1"/>
<font color="Maroon" title="c7">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#E4:44_1"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="rusub_5.html#E6:44_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>;<br/>

<a NAME="E9:44_1"/>
 <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c7">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">v</font>,<font color="Maroon" title="c5">p</font>
 
<b>by </b><i><a class="txt" href="rusub_5.html#E7:44_1"><i><font color="Green" title="E33">A6</font></i></a>, <a class="ref" href="rusub_5.html#T33" target="_self" title="RUSUB_5:th.33">Th33</a></i>;<br/>
<a NAME="E10:44_1"/><b>then </b><i><font color="Green" title="E34">A7</font></i>: 
 <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c7">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="txt" href="rusub_5.html#E4:44_1"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:44_1"/>
( <font color="Maroon" title="c7">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r</font> &amp; <font color="Maroon" title="c7">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#E4:44_1"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="rusub_5.html#E6:44_1"><i><font color="Green" title="E32">A5</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="E12:44_1"/><b>then </b>
 <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c7">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">v</font>,<font color="Maroon" title="c6">r</font>
 
<b>by </b><i><a class="ref" href="rusub_5.html#T33" target="_self" title="RUSUB_5:th.33">Th33</a></i>;<br/>
<a NAME="E13:44_1"/><b>then </b>
 <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c7">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">N</font>
 
<b>by </b><i><a class="txt" href="rusub_5.html#E6:44_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E14:44_1"/>
 <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c7">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">M</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">N</font>
 <b>by </b><i><a class="txt" href="rusub_5.html#E10:44_1"><i><font color="Green" title="E34">A7</font></i></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:44"/>
<font color="Maroon" title="c2">M</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rusub_5.html#K4" title="RUSUB_5:func.4">Family_open_set</a> <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="ref" href="rusub_5.html#D5" target="_self" title="RUSUB_5:def.5">Def5</a></i>;<br/>


</div>
