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<span class="kw">let </span><font color="Maroon" title="c1">r</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</a> <br/>  for <font color="Olive" title="b2">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">M</font> holds   <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Olive" title="b2">p</font>,<font color="Maroon" title="c1">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Olive" title="b2">p</font>,<font color="Maroon" title="c1">r</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">M</font> be    <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</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">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font> holds   <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">r</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">r</font>)</span><br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:45_1"/>
(  not <font color="Maroon" title="c2">M</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  or <font color="Maroon" title="c2">M</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1"><span class="kw">suppose </span></a><a NAME="E1:45_1_1"/>
 not <font color="Maroon" title="c2">M</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">r</font>)</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">M9</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</a> , <font color="Maroon" title="c5">p9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">M9</font><span class="kw"> such that </span><br/><a NAME="E3:45_1_1"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
( <font color="Maroon" title="c4">M9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c5">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> )
 <span class="kw">and </span><br/><a NAME="E4:45_1_1"/>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">r</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">M9</font> :  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Maroon" title="c5">p9</font>,<font color="Olive" title="b1">q</font>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">r</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D14" target="_self" title="METRIC_1:def.14">Def15</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">M9</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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c5">p9</font>,<font color="Maroon" title="c1">r</font>) implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c5">p9</font>,<font color="Maroon" title="c1">r</font>) )</span><br/><span class="kw">assume </span><a NAME="E1:45_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c5">p9</font>,<font color="Maroon" title="c1">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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c5">p9</font>,<font color="Maroon" title="c1">r</font>)</span><br/><a NAME="E2:45_1_1_1"/><span class="kw">then </span>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Maroon" title="c5">p9</font>,<font color="Maroon" title="c6">x</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T11" target="_self" title="METRIC_1:th.11">Th12</a></span>;<br/><a NAME="E3:45_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">M9</font> :  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Maroon" title="c5">p9</font>,<font color="Olive" title="b1">q</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">r</font> </span> } </span> 
 ;<br/><span class="kw">hence </span><a NAME="E4:45_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c5">p9</font>,<font color="Maroon" title="c1">r</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="metric_1.html#E32"><span class="lab"><font color="Green" title="E29">Lm6</font></span></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">hence </span><a NAME="E6:45_1_1"/>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">r</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_1_1"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="ref" href="subset_1.html#T2" title="SUBSET_1:th.2">SUBSET_1:2</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2"><span class="kw">suppose </span></a><a NAME="E1:45_1_2"/><span class="lab"><font color="Green" title="E35">A2</font></span>: 
<font color="Maroon" title="c2">M</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">r</font>)</span><br/><div class="add"><a NAME="E2:45_1_2"/><span class="kw">then </span>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">r</font>) is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D14" target="_self" title="METRIC_1:def.14">Def15</a></span>;<br/><span class="kw">hence </span><a NAME="E3:45_1_2"/>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">r</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1_2"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="metric_1.html#D15" target="_self" title="METRIC_1:def.15">Def16</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></div><span class="kw">end;</span></div>

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