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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  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="c3">B</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="c4">B'</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="c2">A</font></span>)</span>;<br/>







<b>assume </b><a NAME="E1:28"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B'</font>
 ;<br/>

<b>thus </b><a NAME="E2:28"/>
( <font color="Maroon" title="c4">B'</font> is <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> implies <font color="Maroon" title="c3">B</font> is <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_1"/><i><font color="Green" title="E24">A2</font></i>: 
<font color="Maroon" title="c4">B'</font> is <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>
 ;<br/>

<a NAME="E2:28_1"/>
<font color="Maroon" title="c4">B'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c2">A</font></span>)</span>
 
;<br/>
<a NAME="E3:28_1"/><b>then </b>
<font color="Maroon" title="c4">B'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="topmetr.html#D2" title="TOPMETR:def.2">TOPMETR:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:28_1"/>
<font color="Maroon" title="c3">B</font> is <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>
 <b>by </b><i><a class="txt" href="compl_sp.html#E1:28"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="compl_sp.html#E1:28_1"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="hausdorf.html#T19" title="HAUSDORF:th.19">HAUSDORF:19</a></i>;<br/>


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

<b>assume </b><a NAME="E3:28"/><i><font color="Green" title="E24">A3</font></i>: 
<font color="Maroon" title="c3">B</font> is <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_2"/>
( <font color="Maroon" title="c4">B'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <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="c2">A</font></span>)</span> or <font color="Maroon" title="c4">B'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <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="c2">A</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1"><b>suppose </b></a><a NAME="E1:28_2_1"/>
<font color="Maroon" title="c4">B'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <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="c2">A</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:28_2_1"/>
<font color="Maroon" title="c4">B'</font> is <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_2"><b>suppose </b></a><a NAME="E1:28_2_2"/>
<font color="Maroon" title="c4">B'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <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="c2">A</font></span>)</span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:28_2_2"/><i><font color="Green" title="E25">A4</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B'</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">p</font> = <font color="Maroon" title="c5">p</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="c2">A</font></span>)</span> <b>by </b><i><a class="txt" href="compl_sp.html#E3:28_2_2"><i><font color="Green" title="E25">A4</font></i></a></i>;<br/><a NAME="E5:28_2_2"/><i><font color="Green" title="E26">A5</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <b>by </b><i><a class="txt" href="compl_sp.html#E3:28"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="tbsp_1.html#T29" title="TBSP_1:th.29">TBSP_1:29</a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">q</font> be   <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="c2">A</font></span>)</span>;<br/><b>assume </b><a NAME="E1:28_2_2_1"/><i><font color="Green" title="E27">A6</font></i>: 
<font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B'</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c8">p'</font> = <font color="Maroon" title="c6">p</font>, <font color="Maroon" title="c9">q'</font> = <font color="Maroon" title="c7">q</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> <b>by </b><i><a class="ref" href="topmetr.html#T12" title="TOPMETR:th.12">TOPMETR:12</a></i>;<br/><a NAME="E3:28_2_2_1"/><i><font color="Green" title="E28">A7</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</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">p'</font>,<font color="Maroon" title="c9">q'</font>
 <b>by </b><i><a class="ref" href="topmetr.html#D1" title="TOPMETR:def.1">TOPMETR:def 1</a></i>;<br/><a NAME="E4:28_2_2_1"/>
(  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">p'</font>,<font color="Maroon" title="c9">q'</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="c3">B</font> &amp; <span class="p1">(<span class="default"><a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 )
 <b>by </b><i><a class="txt" href="compl_sp.html#E1:28"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="compl_sp.html#E3:28"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="compl_sp.html#E3:28_2_2"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="compl_sp.html#E1:28_2_2_1"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="tbsp_1.html#D10" title="TBSP_1:def.10">TBSP_1:def 10</a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E5:28_2_2_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="tbsp_1.html#K2" title="TBSP_1:func.2">diameter</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <b>by </b><i><a class="txt" href="compl_sp.html#E3:28_2_2_1"><i><font color="Green" title="E28">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:28_2_2"/>
<font color="Maroon" title="c4">B'</font> is <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a>
 <b>by </b><i><a class="txt" href="compl_sp.html#E3:28_2_2"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="compl_sp.html#E5:28_2_2"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="tbsp_1.html#T15" title="TBSP_1:th.15">TBSP_1:15</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
