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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="metric_1.html#V6">Reflexive</a>  <a href="metric_1.html#L1">MetrStruct</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:40"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="tbsp_1.html#V6" target="_self">bounded</a>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1"><b>suppose </b></a><a NAME="E1:40_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:40_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="tbsp_1.html#E1:40"><i><font color="Green">E27</font></i></a>, <a class="ref" href="tbsp_1.html#D10" target="_self">Def10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2"><b>suppose </b></a><a NAME="E1:40_1_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="tbsp_1.html#E1:40_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E4:40_1_2"/>
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="tbsp_1.html#E1:40"><i><font color="Green">E27</font></i></a>, <a class="txt" href="tbsp_1.html#E1:40_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tbsp_1.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E5:40_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="metric_1.html#T1">METRIC_1:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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