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<b>let </b><font color="Maroon" title="c1">PM</font> be    <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</a> ;<br/><b>let </b><font color="Maroon" title="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">PM</font>;<br/><b>let </b><font color="Maroon" title="c3">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>assume </b><a NAME="E1:34"/>
<font color="Maroon" title="c1">PM</font> is <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>
 ;<br/>

<a NAME="E2:34"/><b>then </b>
for <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">PM</font>  st <font color="Olive" title="b1">y</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="c2">x</font>,<font color="Maroon" title="c3">r</font> holds <br/> ex <font color="Olive" title="b2">p</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font> )
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T30" target="_self" title="PCOMPS_1:th.30">Th30</a></i>;<br/>
<b>hence </b><a NAME="E3:34"/>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">Family_open_set</a> <font color="Maroon" title="c1">PM</font>
 <b>by </b><i><a class="ref" href="pcomps_1.html#D5" target="_self" title="PCOMPS_1:def.5">Def5</a></i>;<br/>


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