<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</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><br/>  for <font color="Olive" title="b2">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><br/>  for <font color="Olive" title="b3">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bhsp_2.html#K6" title="BHSP_2:func.6">Sphere</a> (<font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">r</font>) holds <br/><font color="Olive" title="b1">M</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> </span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><br/>  for <font color="Olive" title="b2">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bhsp_2.html#K6" title="BHSP_2:func.6">Sphere</a> (<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">r</font>) holds <br/><font color="Maroon" title="c2">M</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> </span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bhsp_2.html#K6" title="BHSP_2:func.6">Sphere</a> (<font color="Maroon" title="c3">v</font>,<font color="Olive" title="b1">r</font>) holds <br/><font color="Maroon" title="c2">M</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bhsp_2.html#K6" title="BHSP_2:func.6">Sphere</a> (<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">r</font>) implies <font color="Maroon" title="c2">M</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  )</span><br/>







<span class="kw">reconsider </span><font color="Maroon" title="c5">B</font> =  <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>), <font color="Maroon" title="c6">C</font> =  <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> (<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">r</font>) 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/>
<span class="kw">assume </span><a NAME="E2:60"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
<font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bhsp_2.html#K6" title="BHSP_2:func.6">Sphere</a> (<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">r</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">M</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> </span><br/>

<a NAME="E3:60"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
<font color="Maroon" title="c2">M</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><span class="kw">proof </span></a><div class="add">

<div><span class="kw">hereby </span> <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>,<a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">C</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> </span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c7">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">a</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 <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">C</font> )</span><br/><span class="kw">assume </span><a NAME="E1:60_1_1"/><span class="lab"><font color="Green" title="E34">A3</font></span>: 
<font color="Maroon" title="c7">a</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">C</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">e</font> = <font color="Maroon" title="c7">a</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"/>
 not <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_1_1"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E4:60_1_1"/><span class="kw">then </span>
 <a href="bhsp_1.html#K4" title="BHSP_1:func.4">dist</a> (<font color="Maroon" title="c8">e</font>,<font color="Maroon" title="c3">v</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="bhsp_2.html#T52" title="BHSP_2:th.52">BHSP_2:52</a></span>;<br/><a NAME="E5:60_1_1"/><span class="kw">then </span>
(  <a href="bhsp_1.html#K4" title="BHSP_1:func.4">dist</a> (<font color="Maroon" title="c8">e</font>,<font color="Maroon" title="c3">v</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> or  <a href="bhsp_1.html#K4" title="BHSP_1:func.4">dist</a> (<font color="Maroon" title="c8">e</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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E6:60_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c8">e</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="c3">v</font>,<font color="Maroon" title="c4">r</font>) or  not <font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_2.html#T41" title="BHSP_2:th.41">BHSP_2:41</a>, <a class="ref" href="bhsp_2.html#T48" title="BHSP_2:th.48">BHSP_2:48</a></span>;<br/><a NAME="E7:60_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c8">e</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="c3">v</font>,<font color="Maroon" title="c4">r</font>) or <font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><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>)</span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T29" title="SUBSET_1:th.29">SUBSET_1:29</a></span>;<br/><span class="kw">hence </span><a NAME="E8:60_1_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c7">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">C</font> or <font color="Maroon" title="c7">a</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>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:60_1"/><span class="lab"><font color="Green" title="E34">A4</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">a</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> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">e</font> = <font color="Maroon" title="c7">a</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="E4:60_1"/>
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60_1"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E5:60_1"/><span class="kw">then </span>
(  not <font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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>) or <font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E6:60_1"/><span class="kw">then </span>
(  <a href="bhsp_1.html#K4" title="BHSP_1:func.4">dist</a> (<font color="Maroon" title="c8">e</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> or  <a href="bhsp_1.html#K4" title="BHSP_1:func.4">dist</a> (<font color="Maroon" title="c8">e</font>,<font color="Maroon" title="c3">v</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_2.html#T41" title="BHSP_2:th.41">BHSP_2:41</a>, <a class="ref" href="bhsp_2.html#T48" title="BHSP_2:th.48">BHSP_2:48</a></span>;<br/>
<a NAME="E7:60_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="bhsp_2.html#T52" title="BHSP_2:th.52">BHSP_2:52</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:60_1"/>
<font color="Maroon" title="c7">a</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60_1"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="ref" href="subset_1.html#T29" title="SUBSET_1:th.29">SUBSET_1:29</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:60"/>
( <font color="Maroon" title="c5">B</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  &amp; <font color="Maroon" title="c6">C</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rusub_5.html#T50" target="_self" title="RUSUB_5:th.50">Th50</a>, <a class="ref" href="rusub_5.html#T53" target="_self" title="RUSUB_5:th.53">Th53</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:60"/>
<font color="Maroon" title="c2">M</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="tops_1.html#T3" title="TOPS_1:th.3">TOPS_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
