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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/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_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="41_1_1"><b>suppose </b></a><a NAME="E1:41_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="E3:41_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="E4:41_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="41_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="E1:41_1_1_1"/>
<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>
 ;<br/><a NAME="E2:41_1_1_1"/><b>then </b>
 <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>
 <b>by </b><i><a class="ref" href="metric_1.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E3:41_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> </span>}</span> 
 ;<br/><b>hence </b><a NAME="E4:41_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>
 <b>by </b><i><a class="txt" href="metric_1.html#E7"><i><font color="Green">Lemma27</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:41_1_1"/>
 <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>
 <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="41_1_2"><b>suppose </b></a><a NAME="E1:41_1_2"/>
<font color="Maroon">M</font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><div class="add"><a NAME="E2:41_1_2"/><b>then </b>
(  <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> &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> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:41_1_2"/>
 <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>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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