<?xml version="1.0"?>
<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> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="rusub_5.html#K5" title="RUSUB_5:func.5">TopUnitSpace</a> <font color="Maroon" title="c1">V</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c3">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>;<br/><b>let </b><font color="Maroon" title="c4">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>







<b>assume </b><a NAME="E1:60"/><i><font color="Green" title="E32">A1</font></i>: 
<font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bhsp_2.html#K5" title="BHSP_2:func.5">cl_Ball</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">r</font>
 ;<br/>

<a NAME="E2:60"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  iff  ex <font color="Olive" title="b2">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="rusub_5.html#K5" title="RUSUB_5:func.5">TopUnitSpace</a> <font color="Maroon" title="c1">V</font></span>)</span> st <br/>( <font color="Olive" title="b2">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <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#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">Q</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>thus </b><a NAME="E1:60_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  implies  ex <font color="Olive" title="b1">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="rusub_5.html#K5" title="RUSUB_5:func.5">TopUnitSpace</a> <font color="Maroon" title="c1">V</font></span>)</span> st <br/>( <font color="Olive" title="b1">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b1">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#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Q</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:60_1_1"/><i><font color="Green" title="E33">A2</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">e</font> = <font color="Maroon" title="c5">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">Q</font> =  <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c6">e</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">v</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">r</font></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="rusub_5.html#K5" title="RUSUB_5:func.5">TopUnitSpace</a> <font color="Maroon" title="c1">V</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon" title="c7">Q</font>
;<br/>

<b>thus </b><a NAME="E4:60_1_1"/>
<font color="Maroon" title="c7">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="rusub_5.html#T50" target="_self" title="RUSUB_5:th.50">Th50</a></i>;<br/>

<b>thus </b><a NAME="E5:60_1_1"/>
<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#K3" title="SUBSET_1:func.3">`</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:60_1_1_1"/><i><font color="Green" title="E34">A3</font></i>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Q</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c9">f</font> = <font color="Maroon" title="c8">q</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<a NAME="E3:60_1_1_1"/><i><font color="Green" title="E35">A4</font></i>: 
 <a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c9">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">v</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">r</font>
 
<b>by </b><i><a class="txt" href="rusub_5.html#E1:60_1_1_1"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="bhsp_2.html#T41" title="BHSP_2:th.41">BHSP_2:41</a></i>;<br/>
<a NAME="E4:60_1_1_1"/>
 <a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">v</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c9">f</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c9">f</font>,<font color="Maroon" title="c3">v</font></span>)</span>
 
<b>by </b><i><a class="ref" href="bhsp_1.html#T42" title="BHSP_1:th.42">BHSP_1:42</a></i>;<br/>
<a NAME="E5:60_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">v</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c9">f</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c9">f</font>,<font color="Maroon" title="c3">v</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">r</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E6:60_1_1_1"/><b>then </b>
 <a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c9">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c9">f</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c9">f</font>,<font color="Maroon" title="c3">v</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">r</font>
 
<b>by </b><i><a class="txt" href="rusub_5.html#E3:60_1_1_1"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:60_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c9">f</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c9">f</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c9">f</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c9">f</font>,<font color="Maroon" title="c3">v</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">r</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c9">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E8:60_1_1_1"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c9">f</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c9">f</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c9">f</font>,<font color="Maroon" title="c3">v</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">r</font>
 
;<br/>
<a NAME="E9:60_1_1_1"/><b>then </b>
 <a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c9">f</font>,<font color="Maroon" title="c3">v</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">r</font>
 
<b>by </b><i><a class="ref" href="real_2.html#T106" title="REAL_2:th.106">REAL_2:106</a></i>;<br/>
<a NAME="E10:60_1_1_1"/><b>then </b>
not <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="txt" href="rusub_5.html#E1:60"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="bhsp_2.html#T48" title="BHSP_2:th.48">BHSP_2:48</a></i>;<br/>
<b>hence </b><a NAME="E11:60_1_1_1"/>
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="rusub_5.html#E1:60_1_1_1"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="subset_1.html#T50" title="SUBSET_1:th.50">SUBSET_1:50</a></i>;<br/>


</div><b>end;</b></div>

<a NAME="E6:60_1_1"/>
not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="txt" href="rusub_5.html#E1:60_1_1"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E7:60_1_1"/><b>then </b>
 <a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c6">e</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">r</font>
 
<b>by </b><i><a class="txt" href="rusub_5.html#E1:60"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="bhsp_2.html#T48" title="BHSP_2:th.48">BHSP_2:48</a></i>;<br/>
<a NAME="E8:60_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="bhsp_1.html#K5" title="BHSP_1:func.5">dist</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">v</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">r</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<b>hence </b><a NAME="E9:60_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Q</font>
 <b>by </b><i><a class="ref" href="bhsp_2.html#T42" title="BHSP_2:th.42">BHSP_2:42</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:60_1"/>
(  ex <font color="Olive" title="b1">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="rusub_5.html#K5" title="RUSUB_5:func.5">TopUnitSpace</a> <font color="Maroon" title="c1">V</font></span>)</span> st <br/>( <font color="Olive" title="b1">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b1">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#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Q</font> ) implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:60"/><b>then </b>
<font color="Maroon" title="c2">M</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T57" title="TOPS_1:th.57">TOPS_1:57</a></i>;<br/>
<b>hence </b><a NAME="E4:60"/>
<font color="Maroon" title="c2">M</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T29" title="TOPS_1:th.29">TOPS_1:29</a></i>;<br/>


</div>
