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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> ;<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="E1:61"/><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0
 ;<br/>

<b>thus </b><a NAME="E2:61"/>
 <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="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
  <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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:61_1"/><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c4">a</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="c3">z</font>,<font color="Maroon" title="c1">r</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">b</font> = <font color="Maroon" title="c4">a</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">M</font> ;<br/>
<a NAME="E3:61_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">r</font>
 
<b>by </b><i><a class="txt" href="topreal6.html#E1:61_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="metric_1.html#T14" title="METRIC_1:th.14">METRIC_1:14</a></i>;<br/>
<b>hence </b><a NAME="E4:61_1"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="topreal6.html#E1:61"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="metric_1.html#T5" title="METRIC_1:th.5">METRIC_1:5</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:61"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/>


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