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<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> ;<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> ;<br/><b>let </b><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font>;<br/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:38_1"/><i><font color="Green" title="E31">A1</font></i>: 
<font color="Maroon" title="c4">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="c3">p</font>,<font color="Maroon" title="c1">r</font>
 ;<br/><a NAME="E2:38_1"/><i><font color="Green" title="E32">A2</font></i>: 
not <font color="Maroon" title="c2">M</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:38_1_1"/>
not the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="metric_1.html#E1:38_1"><i><font color="Green" title="E31">A1</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a><br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c5">M'</font> = <font color="Maroon" title="c2">M</font> as  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>  ;<br/><b>reconsider </b><font color="Maroon" title="c6">p'</font> = <font color="Maroon" title="c3">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">M'</font> ;<br/><a NAME="E5:38_1"/>
<font color="Maroon" title="c4">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="c5">M'</font> :  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c6">p'</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> 
 <b>by </b><i><a class="txt" href="metric_1.html#E1:38_1"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="metric_1.html#E35"><i><font color="Green" title="E27">Lm7</font></i></a></i>;<br/><a NAME="E6:38_1"/><b>then </b>
 ex <font color="Olive" title="b1">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font> st <br/>( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">p</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> )
 ;<br/><b>hence </b><a NAME="E7:38_1"/>
( not <font color="Maroon" title="c2">M</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">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="txt" href="metric_1.html#E2:38_1"><i><font color="Green" title="E32">A2</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:38"/>
not <font color="Maroon" title="c2">M</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">M'</font> = <font color="Maroon" title="c2">M</font> as  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>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c6">p'</font> = <font color="Maroon" title="c3">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">M'</font> ;<br/>
<b>assume </b><a NAME="E5:38"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">r</font>
 ;<br/>

<a NAME="E6:38"/><b>then </b>
<font color="Maroon" title="c4">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="c5">M'</font> :  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c6">p'</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/>
<b>hence </b><a NAME="E7:38"/>
<font color="Maroon" title="c4">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="c3">p</font>,<font color="Maroon" title="c1">r</font>
 <b>by </b><i><a class="txt" href="metric_1.html#E35"><i><font color="Green" title="E27">Lm7</font></i></a></i>;<br/>


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