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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>, <font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>;<br/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:38_1"/><i><font color="Green">E39</font></i>: 
<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="E2:38_1"/><i><font color="Green">E40</font></i>: 
not <font color="Maroon">M</font> is <a href="struct_0.html#V3">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">carrier</a> of <font color="Maroon">M</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="metric_1.html#D5" target="_self">Def5</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a><br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">M'</font> = <font color="Maroon">M</font> as  non <a href="struct_0.html#V3">empty</a>  <a href="metric_1.html#L1" target="_self">MetrStruct</a>  ;<br/><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M'</font> ;<br/><a NAME="E5:38_1"/>
<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> 
 <b>by </b><i><a class="ref" href="metric_1.html#D5" target="_self">Def5</a>, <a class="txt" href="metric_1.html#E7"><i><font color="Green">Lemma27</font></i></a></i>;<br/><a NAME="E6:38_1"/><b>then </b>
 ex <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> &amp;  <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> )
 ;<br/><b>hence </b><a NAME="E7:38_1"/>
( not <font color="Maroon">M</font> is <a href="struct_0.html#V3">empty</a> &amp;  <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#D6" target="_self">Def6</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:38"/>
not <font color="Maroon">M</font> is <a href="struct_0.html#V3">empty</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">M'</font> = <font color="Maroon">M</font> as  non <a href="struct_0.html#V3">empty</a>  <a href="metric_1.html#L1" target="_self">MetrStruct</a>  ;<br/>
<b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M'</font> ;<br/>
<b>assume </b><a NAME="E5:38"/>
 <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>
 ;<br/>

<a NAME="E6:38"/><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="E7:38"/>
<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/>


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