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<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">M</font> be  non <a href="struct_0.html#V3">empty</a> <a href="metric_1.html#NM1">MetrSpace</a>;<br/><b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>, <font color="Maroon">B</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:13"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Olive">p</font>,<font color="Olive">q</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 ;<br/>

<b>set </b><font color="Maroon">X</font> =  <a href="weierstr.html#K3">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">A</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">B</font></span>)</span>;<br/>
<a NAME="E2:13"/><i><font color="Green">E40</font></i>: 
 <a href="weierstr.html#K3">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">A</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">A</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="weierstr.html#D3">WEIERSTR:def 3</a></i>;<br/>
<a NAME="E3:13"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">s</font> <a href="hidden.html#R2">in</a>  <a href="weierstr.html#K3">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">A</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">B</font></span>)</span> holds <br/><font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:13_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="weierstr.html#K3">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">A</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">B</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:13_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">A</font></span>)</span>
 <b>and </b><br/><a NAME="E4:13_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E5:13_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">y</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font> <b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<a NAME="E8:13_1"/>
for <font color="Olive">q</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>  st <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Maroon">p</font>,<font color="Olive">q</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E9:13_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:13"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:13"/><b>then </b>
not  <a href="weierstr.html#K3">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">A</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">B</font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T152">RELAT_1:152</a></i>;<br/>
<a NAME="E6:13"/><b>then </b>
 <a href="seq_4.html#K5">lower_bound</a> <span class="p1">(<span class="default"><a href="weierstr.html#K3">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">A</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="seq_4.html#T60">SEQ_4:60</a></i>;<br/>
<a NAME="E7:13"/><b>then </b>
 <a href="weierstr.html#K5">lower_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">A</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">B</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="weierstr.html#D5">WEIERSTR:def 5</a></i>;<br/>
<b>hence </b><a NAME="E8:13"/>
 <a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="weierstr.html#D9">WEIERSTR:def 9</a></i>;<br/>


</div>
