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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>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:43"/><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>
 <b>and </b><br/><a NAME="E2:43"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E3:43"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="tbsp_1.html#V6" target="_self">bounded</a>
 
<b>by </b><i><a class="txt" href="tbsp_1.html#E1:43"><i><font color="Green">E27</font></i></a>, <a class="txt" href="tbsp_1.html#E2:43"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tbsp_1.html#T21" target="_self">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">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>3</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="43_1_1"><b>suppose </b></a><a NAME="E1:43_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:43_1_1"/><b>then </b>
 <a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="tbsp_1.html#E3:43"><i><font color="Green">E29</font></i></a>, <a class="ref" href="tbsp_1.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E3:43_1_1"/>
 <a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">c<sub>3</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#E1:43"><i><font color="Green">E27</font></i></a>, <a class="ref" href="tbsp_1.html#T29" target="_self">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">E30</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:43_1_2"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/> <a href="metric_1.html#K2">dist</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</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#E1:43"><i><font color="Green">E27</font></i></a>, <a class="txt" href="tbsp_1.html#E2:43"><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="E3:43_1_2"/>
 <a href="tbsp_1.html#K2" target="_self">diameter</a> <font color="Maroon">c<sub>3</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#E3:43"><i><font color="Green">E29</font></i></a>, <a class="txt" href="tbsp_1.html#E1:43_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="tbsp_1.html#D10" target="_self">Def10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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