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<div class="add">

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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  <span class="p1">(<span class="default"><a href="metric_1.html#K11" title="METRIC_1:func.11">Sphere</a> <font color="Olive" title="b2">p</font>,<font color="Maroon" title="c1">r</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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  <span class="p1">(<span class="default"><a href="metric_1.html#K11" title="METRIC_1:func.11">Sphere</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">r</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="metric_1.html#K11" title="METRIC_1:func.11">Sphere</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">r</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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/>





<a NAME="E1:46"/>
(  <a href="metric_1.html#K11" title="METRIC_1:func.11">Sphere</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> &amp;  <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> )
 
<b>by </b><i><a class="ref" href="metric_1.html#T15" target="_self" title="METRIC_1:th.15">Th15</a>, <a class="ref" href="metric_1.html#T16" target="_self" title="METRIC_1:th.16">Th16</a></i>;<br/>
<b>hence </b><a NAME="E2:46"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K11" title="METRIC_1:func.11">Sphere</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">r</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><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></span>)</span> <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>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K11" title="METRIC_1:func.11">Sphere</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">r</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><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></span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_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="46_1_1"><b>suppose </b></a><a NAME="E1:46_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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K11" title="METRIC_1:func.11">Sphere</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">r</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><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></span>)</span></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c4">M'</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">p'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">M'</font><b> such that </b><br/><a NAME="E3:46_1_1"/><i><font color="Green" title="E33">A1</font></i>: 
( <font color="Maroon" title="c4">M'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c5">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> )
 <b>and </b><br/><a NAME="E4:46_1_1"/>
 <a href="metric_1.html#K11" title="METRIC_1:func.11">Sphere</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">M'</font> :  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">p'</font>,<font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="metric_1.html#D17" target="_self" title="METRIC_1:def.17">Def17</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_1"><b>now </b></a><div class="add"><b>let </b><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">M'</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">p'</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> <span class="p1">(<span class="default"><a href="metric_1.html#K11" title="METRIC_1:func.11">Sphere</a> <font color="Maroon" title="c5">p'</font>,<font color="Maroon" title="c1">r</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c5">p'</font>,<font color="Maroon" title="c1">r</font></span>)</span> )</span><br/><b>assume </b><a NAME="E1:46_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">p'</font>,<font color="Maroon" title="c1">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="metric_1.html#K11" title="METRIC_1:func.11">Sphere</a> <font color="Maroon" title="c5">p'</font>,<font color="Maroon" title="c1">r</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c5">p'</font>,<font color="Maroon" title="c1">r</font></span>)</span></span><br/><a NAME="E2:46_1_1_1"/><b>then </b><i><font color="Green" title="E34">A2</font></i>: 
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">p'</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>
 <b>by </b><i><a class="ref" href="metric_1.html#T13" target="_self" title="METRIC_1:th.13">Th13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_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="E1:46_1_1_1_1_1"/>
(  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">p'</font>,<font color="Maroon" title="c6">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">r</font> or  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">p'</font>,<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font> )
 </span><b>by </b><i><a class="txt" href="metric_1.html#E2:46_1_1_1"><i><font color="Green" title="E34">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="46_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:46_1_1_1_1_1_1"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">p'</font>,<font color="Maroon" title="c6">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">p'</font>,<font color="Maroon" title="c1">r</font></span><br/><div class="add"><b>hence </b><a NAME="E2:46_1_1_1_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">p'</font>,<font color="Maroon" title="c1">r</font>
 <b>by </b><i><a class="ref" href="metric_1.html#T12" target="_self" title="METRIC_1:th.12">Th12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:46_1_1_1_1_1_2"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">p'</font>,<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#K11" title="METRIC_1:func.11">Sphere</a> <font color="Maroon" title="c5">p'</font>,<font color="Maroon" title="c1">r</font></span><br/><div class="add"><b>hence </b><a NAME="E2:46_1_1_1_1_1_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K11" title="METRIC_1:func.11">Sphere</a> <font color="Maroon" title="c5">p'</font>,<font color="Maroon" title="c1">r</font>
 <b>by </b><i><a class="ref" href="metric_1.html#T14" target="_self" title="METRIC_1:th.14">Th14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:46_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="metric_1.html#K11" title="METRIC_1:func.11">Sphere</a> <font color="Maroon" title="c5">p'</font>,<font color="Maroon" title="c1">r</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c5">p'</font>,<font color="Maroon" title="c1">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E6:46_1_1"/>
 <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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K11" title="METRIC_1:func.11">Sphere</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">r</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><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></span>)</span>
 <b>by </b><i><a class="txt" href="metric_1.html#E3:46_1_1"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_2"><b>suppose </b></a><a NAME="E1:46_1_2"/>
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K11" title="METRIC_1:func.11">Sphere</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">r</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><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></span>)</span></span><br/><div class="add"><a NAME="E2:46_1_2"/><b>then </b>
(  <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> &amp;  <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> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="metric_1.html#K11" title="METRIC_1:func.11">Sphere</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> )
 <b>by </b><i><a class="ref" href="metric_1.html#D15" target="_self" title="METRIC_1:def.15">Def15</a>, <a class="ref" href="metric_1.html#D16" target="_self" title="METRIC_1:def.16">Def16</a>, <a class="ref" href="metric_1.html#D17" target="_self" title="METRIC_1:def.17">Def17</a></i>;<br/><b>hence </b><a NAME="E3:46_1_2"/>
 <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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K11" title="METRIC_1:func.11">Sphere</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c1">r</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><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></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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