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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon" title="c2">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>





<a NAME="E1:62"/><i><font color="Green" title="E36">A1</font></i>: 
<span class="p1">(<span class="default"><a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="bhsp_2.html#K6" title="BHSP_2:func.6">Sphere</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bhsp_2.html#K5" title="BHSP_2:func.5">cl_Ball</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><b>proof </b></a><div class="add">

<a NAME="E1:62_1"/>
(  <a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bhsp_2.html#K5" title="BHSP_2:func.5">cl_Ball</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font> &amp;  <a href="bhsp_2.html#K6" title="BHSP_2:func.6">Sphere</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bhsp_2.html#K5" title="BHSP_2:func.5">cl_Ball</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font> )
 
<b>by </b><i><a class="ref" href="bhsp_2.html#T54" target="_self" title="BHSP_2:th.54">Th54</a>, <a class="ref" href="bhsp_2.html#T55" target="_self" title="BHSP_2:th.55">Th55</a></i>;<br/>
<b>hence </b><a NAME="E2:62_1"/>
<span class="p1">(<span class="default"><a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="bhsp_2.html#K6" title="BHSP_2:func.6">Sphere</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bhsp_2.html#K5" title="BHSP_2:func.5">cl_Ball</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>;<br/><b>assume </b><a NAME="E1:62_2"/>
<font color="Maroon" title="c4">y</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="c2">x</font>,<font color="Maroon" title="c3">r</font>
 ;<br/><a NAME="E2:62_2"/><b>then </b><i><font color="Green" title="E37">A2</font></i>: 
<span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="ref" href="bhsp_2.html#T47" target="_self" title="BHSP_2:th.47">Th47</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:62_2_1_1"/>
( <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font> or <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> )
 </span><b>by </b><i><a class="txt" href="bhsp_2.html#E2:62_2"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_2_1_1_1"><b>case </b></a><a NAME="E1:62_2_1_1_1"/>
<span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">r</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:62_2_1_1_1"/>
<font color="Maroon" title="c4">y</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">x</font>,<font color="Maroon" title="c3">r</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_2_1_1_2"><b>case </b></a><a NAME="E1:62_2_1_1_2"/>
<span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:62_2_1_1_2"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bhsp_2.html#K6" title="BHSP_2:func.6">Sphere</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:62_2"/>
<font color="Maroon" title="c4">y</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">x</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="bhsp_2.html#K6" title="BHSP_2:func.6">Sphere</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:62"/><b>then </b>
 <a href="bhsp_2.html#K5" title="BHSP_2:func.5">cl_Ball</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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">x</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="bhsp_2.html#K6" title="BHSP_2:func.6">Sphere</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></i>;<br/>
<b>hence </b><a NAME="E4:62"/>
<span class="p1">(<span class="default"><a href="bhsp_2.html#K4" title="BHSP_2:func.4">Ball</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="bhsp_2.html#K6" title="BHSP_2:func.6">Sphere</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font></span>)</span> <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="c2">x</font>,<font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="txt" href="bhsp_2.html#E1:62"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


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