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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">D</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<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="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">C</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="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:44"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 ;<br/>

<b>consider </b><font color="Maroon">A'</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span></span>)</span>, <font color="Maroon">D'</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:44"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A'</font>
 <b>and </b><br/><a NAME="E4:44"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D'</font>
 <b>and </b><br/><a NAME="E5:44"/><i><font color="Green">E43</font></i>: 
 <a href="jordan1k.html#K1" target="_self">dist_min</a> <font color="Maroon">A</font>,<font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">A'</font>,<font color="Maroon">D'</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">A''</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span></span>)</span>, <font color="Maroon">C'</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E7:44"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A''</font>
 <b>and </b><br/><a NAME="E8:44"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C'</font>
 <b>and </b><br/><a NAME="E9:44"/><i><font color="Green">E49</font></i>: 
 <a href="jordan1k.html#K1" target="_self">dist_min</a> <font color="Maroon">A</font>,<font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">A''</font>,<font color="Maroon">C'</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">f</font> =  <a href="weierstr.html#K8">dist_min</a> <font color="Maroon">A'</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span></span>)</span>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E11:44"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <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> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">C'</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i>, <a class="ref" href="relat_1.html#T152">RELAT_1:152</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D'</font> as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  ;<br/>
<i><font color="Green">E94</font></i>:  <a href="pscomp_1.html#NK3" target="_self">inf</a> <font color="Maroon">Y</font> = 
 <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">D'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="weierstr.html#D3">WEIERSTR:def 3</a></i>
<br/>.= 
 <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">D'</font></span>)</span>
<b>by </b><i><a class="ref" href="weierstr.html#D5">WEIERSTR:def 5</a></i>
<br/>.= 
 <a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">A'</font>,<font color="Maroon">D'</font>
<b>by </b><i><a class="ref" href="weierstr.html#D9">WEIERSTR:def 9</a></i>
;<br/>
<i><font color="Green">E95</font></i>:  <a href="pscomp_1.html#NK3" target="_self">inf</a> <font color="Maroon">X</font> = 
 <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">C'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="weierstr.html#D3">WEIERSTR:def 3</a></i>
<br/>.= 
 <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">C'</font></span>)</span>
<b>by </b><i><a class="ref" href="weierstr.html#D5">WEIERSTR:def 5</a></i>
<br/>.= 
 <a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">A'</font>,<font color="Maroon">C'</font>
<b>by </b><i><a class="ref" href="weierstr.html#D9">WEIERSTR:def 9</a></i>
;<br/>
<a NAME="E16:44"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">Y</font> is <a href="seq_4.html#V2">bounded_below</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_3"><b>proof </b></a><div class="add">

<b>take </b>
0
; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_4.html#D2">SEQ_4:def 2</a><br/>

<b>let </b><font color="Maroon">r</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="E1:44_3"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E2:44_3"/><b>then </b>
 ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span></span>)</span> st <br/>( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D'</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<b>hence </b><a NAME="E3:44_3"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:44"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<b>hence </b><a NAME="E18:44"/>
 <a href="jordan1k.html#K1" target="_self">dist_min</a> <font color="Maroon">A</font>,<font color="Maroon">D</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan1k.html#K1" target="_self">dist_min</a> <font color="Maroon">A</font>,<font color="Maroon">C</font>
 <b>by </b><i>, <a class="txt" href="jordan1k.html#E1:44"><i><font color="Green">E36</font></i></a>, <a class="ref" href="jordan1k.html#T3" target="_self">Th3</a>, , , <a class="txt" href="jordan1k.html#E3:44"><i><font color="Green">E40</font></i></a>, , <a class="txt" href="jordan1k.html#E4:44"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan1k.html#E5:44"><i><font color="Green">E43</font></i></a>, <a class="ref" href="seq_4.html#T64">SEQ_4:64</a></i>;<br/>


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