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<b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">M</font> be    <a href="metric_1.html#L1" target="_self">MetrStruct</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>;<br/>





<a NAME="E1:42"/>
(  <a href="metric_1.html#K11" target="_self">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K10" target="_self">cl_Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> &amp;  <a href="metric_1.html#K9" target="_self">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K10" target="_self">cl_Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> )
 
<b>by </b><i><a class="ref" href="metric_1.html#D6" target="_self">Def6</a>, </i>;<br/>
<b>hence </b><a NAME="E2:42"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K11" target="_self">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9" target="_self">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K10" target="_self">cl_Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1"/>
(  not <font color="Maroon">M</font> is <a href="struct_0.html#V3">empty</a> or <font color="Maroon">M</font> is <a href="struct_0.html#V3">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1"><b>suppose </b></a><a NAME="E1:42_1_1"/>
not <font color="Maroon">M</font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">M'</font> being  non <a href="struct_0.html#V3">empty</a>  <a href="metric_1.html#L1" target="_self">MetrStruct</a> , <font color="Maroon">p'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M'</font><b> such that </b><br/><a NAME="E3:42_1_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">M'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> &amp; <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 <b>and </b><br/><a NAME="E4:42_1_1"/>
 <a href="metric_1.html#K11" target="_self">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M'</font> :  <a href="metric_1.html#K2" target="_self">dist</a> <font color="Maroon">p'</font>,<font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> </span>}</span> 
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M'</font>;<br/><b>assume </b><a NAME="E1:42_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K10" target="_self">cl_Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font>
 ;<br/><a NAME="E2:42_1_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="metric_1.html#K2" target="_self">dist</a> <font color="Maroon">p'</font>,<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D4" target="_self">Def4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1_1_1_1_1"/>
(  <a href="metric_1.html#K2" target="_self">dist</a> <font color="Maroon">p'</font>,<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> or  <a href="metric_1.html#K2" target="_self">dist</a> <font color="Maroon">p'</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> )
 </span><b>by </b><i><a class="ref" href="metric_1.html#D8" target="_self">Def8</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:42_1_1_1_1_1_1"/>
 <a href="metric_1.html#K2" target="_self">dist</a> <font color="Maroon">p'</font>,<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9" target="_self">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D3" target="_self">Def3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:42_1_1_1_1_1_2"/>
 <a href="metric_1.html#K2" target="_self">dist</a> <font color="Maroon">p'</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_1_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K11" target="_self">Sphere</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D5" target="_self">Def5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:42_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="metric_1.html#K11" target="_self">Sphere</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9" target="_self">Ball</a> <font color="Maroon">p'</font>,<font color="Maroon">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:42_1_1"/>
 <a href="metric_1.html#K10" target="_self">cl_Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K11" target="_self">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9" target="_self">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="metric_1.html#D7" target="_self">Def7</a>, <a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2"><b>suppose </b></a><a NAME="E1:42_1_2"/>
<font color="Maroon">M</font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><div class="add"><a NAME="E2:42_1_2"/><b>then </b>
(  <a href="metric_1.html#K9" target="_self">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="metric_1.html#K10" target="_self">cl_Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="metric_1.html#K11" target="_self">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> is <a href="xboole_0.html#V1">empty</a> )
 <b>by </b><i><a class="txt" href="metric_1.html#E5"><i><font color="Green">Lemma23</font></i></a>, , </i>;<br/><b>hence </b><a NAME="E3:42_1_2"/>
 <a href="metric_1.html#K10" target="_self">cl_Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K11" target="_self">Sphere</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9" target="_self">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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