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<b>let </b><font color="Maroon" title="c1">M</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font>;<br/><b>let </b><font color="Maroon" title="c3">B</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font>;<br/><b>let </b><font color="Maroon" title="c4">C</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c3">B</font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:18"/><i><font color="Green" title="E18">A1</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c4">C</font> is <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">r0</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:18"/><i><font color="Green" title="E19">A2</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r0</font> &amp; ( 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 <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c3">B</font></span>)</span>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> holds <br/> <a href="metric_1.html#K4" title="METRIC_1:func.4">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> <font color="Maroon" title="c5">r0</font> ) )
 <b>by </b><i><a class="ref" href="tbsp_1.html#D9" title="TBSP_1:def.9">TBSP_1:def 9</a></i>;<br/>
<a NAME="E4:18"/>
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">M</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> holds <br/> <a href="metric_1.html#K4" title="METRIC_1:func.4">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> <font color="Maroon" title="c5">r0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>;<br/>

<b>assume </b><a NAME="E1:18_1"/><i><font color="Green" title="E20">A3</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c8">x0</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c9">y0</font> = <font color="Maroon" title="c7">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c3">B</font></span>)</span> <b>by </b><i><a class="txt" href="hausdorf.html#E1:18"><i><font color="Green" title="E18">A1</font></i></a></i>;<br/>
<a NAME="E3:18_1"/><i><font color="Green" title="E21">A4</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">x0</font>,<font color="Maroon" title="c9">y0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r0</font>
 
<b>by </b><i><a class="txt" href="hausdorf.html#E1:18"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="hausdorf.html#E3:18"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="hausdorf.html#E1:18_1"><i><font color="Green" title="E20">A3</font></i></a></i>;<br/>
<a NAME="E4:18_1"/><i><font color="Green" title="E22">A5</font></i>: 
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c8">x0</font>,<font color="Maroon" title="c9">y0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font>
 
<b>by </b><i><a class="ref" href="topmetr.html#D1" title="TOPMETR:def.1">TOPMETR:def 1</a></i>;<br/>
<a NAME="E5:18_1"/>
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="topmetr.html#K1" title="TOPMETR:func.1">|</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c8">x0</font>,<font color="Maroon" title="c9">y0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">x0</font>,<font color="Maroon" title="c9">y0</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:18_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r0</font>
 <b>by </b><i><a class="txt" href="hausdorf.html#E3:18_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="hausdorf.html#E4:18_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:18"/>
<font color="Maroon" title="c2">A</font> is <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>
 <b>by </b><i><a class="txt" href="hausdorf.html#E3:18"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="tbsp_1.html#D9" title="TBSP_1:def.9">TBSP_1:def 9</a></i>;<br/>


</div>
