<?xml version="1.0"?>
<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> ;<br/><b>let </b><font color="Maroon" title="c2">M</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a> <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a> <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>  <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</a> ;<br/><b>let </b><font color="Maroon" title="c3">z</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">M</font>;<br/>





<b>assume </b><a NAME="E1:63"/>
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<a NAME="E2:63"/><b>then </b><i><font color="Green" title="E27">A1</font></i>: 
(  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="metric_1.html#K11" title="METRIC_1:func.11">Sphere</a> <font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="topreal6.html#T61" target="_self" title="TOPREAL6:th.61">Th61</a>, <a class="ref" href="tbsp_1.html#T17" title="TBSP_1:th.17">TBSP_1:17</a></i>;<br/>
<a NAME="E3:63"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K11" title="METRIC_1:func.11">Sphere</a> <font color="Maroon" title="c3">z</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">z</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">z</font>,<font color="Maroon" title="c1">r</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T17" title="METRIC_1:th.17">METRIC_1:17</a></i>;<br/>
<b>hence </b><a NAME="E4:63"/>
 <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> <font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="topreal6.html#E2:63"><i><font color="Green" title="E27">A1</font></i></a></i>;<br/>


</div>
