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<b>let </b><font color="Maroon" title="c1">R</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#L1" title="METRIC_1:struct.1">MetrStruct</a> ;<br/><b>let </b><font color="Maroon" title="c2">S</font>, <font color="Maroon" title="c3">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:43"/><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon" title="c2">S</font> is <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>
 <b>and </b><br/><a NAME="E2:43"/><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c3">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">S</font>
 ;<br/>

<a NAME="E3:43"/><i><font color="Green" title="E29">A3</font></i>: 
<font color="Maroon" title="c3">V</font> is <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>
 
<b>by </b><i><a class="txt" href="tbsp_1.html#E1:43"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="tbsp_1.html#E2:43"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="tbsp_1.html#T21" target="_self" title="TBSP_1:th.21">Th21</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_1"/>
( <font color="Maroon" title="c3">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c3">V</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1"><b>suppose </b></a><a NAME="E1:43_1_1"/>
<font color="Maroon" title="c3">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:43_1_1"/><b>then </b>
 <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c3">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="tbsp_1.html#E3:43"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="tbsp_1.html#D10" target="_self" title="TBSP_1:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E3:43_1_1"/>
 <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c3">V</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="tbsp_1.html#E1:43"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="tbsp_1.html#T29" target="_self" title="TBSP_1:th.29">Th29</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_2"><b>suppose </b></a><a NAME="E1:43_1_2"/><i><font color="Green" title="E30">A4</font></i>: 
<font color="Maroon" title="c3">V</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:43_1_2"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> holds <br/> <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="tbsp_1.html#E1:43"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="tbsp_1.html#E2:43"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="tbsp_1.html#D10" target="_self" title="TBSP_1:def.10">Def10</a></i>;<br/><b>hence </b><a NAME="E3:43_1_2"/>
 <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c3">V</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="tbsp_1.html#E3:43"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="tbsp_1.html#E1:43_1_2"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="tbsp_1.html#D10" target="_self" title="TBSP_1:def.10">Def10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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