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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">D</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><br/> for <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">C</font> being 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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">D</font> holds <br/> <a href="jordan1k.html#K1" title="JORDAN1K:func.1">dist_min</a> <font color="Olive" title="b2">A</font>,<font color="Olive" title="b1">D</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jordan1k.html#K1" title="JORDAN1K:func.1">dist_min</a> <font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">C</font></span><br/><b>let </b><font color="Maroon" title="c2">D</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">C</font> being 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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">D</font> holds <br/> <a href="jordan1k.html#K1" title="JORDAN1K:func.1">dist_min</a> <font color="Olive" title="b1">A</font>,<font color="Maroon" title="c2">D</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jordan1k.html#K1" title="JORDAN1K:func.1">dist_min</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">C</font></span><br/><b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">C</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">D</font> implies  <a href="jordan1k.html#K1" title="JORDAN1K:func.1">dist_min</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">D</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jordan1k.html#K1" title="JORDAN1K:func.1">dist_min</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">C</font> )</span><br/>





<b>assume </b><a NAME="E1:44"/><i><font color="Green" title="E34">A1</font></i>: 
<font color="Maroon" title="c4">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="jordan1k.html#K1" title="JORDAN1K:func.1">dist_min</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">D</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jordan1k.html#K1" title="JORDAN1K:func.1">dist_min</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">C</font></span><br/>

<b>consider </b><font color="Maroon" title="c5">A'</font>, <font color="Maroon" title="c6">D'</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:44"/><i><font color="Green" title="E35">A2</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A'</font>
 <b>and </b><br/><a NAME="E4:44"/><i><font color="Green" title="E36">A3</font></i>: 
<font color="Maroon" title="c2">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">D'</font>
 <b>and </b><br/><a NAME="E5:44"/><i><font color="Green" title="E37">A4</font></i>: 
 <a href="jordan1k.html#K1" title="JORDAN1K:func.1">dist_min</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c5">A'</font>,<font color="Maroon" title="c6">D'</font>
 <b>by </b><i><a class="ref" href="jordan1k.html#D1" target="_self" title="JORDAN1K:def.1">Def1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">A''</font>, <font color="Maroon" title="c8">C'</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E7:44"/><i><font color="Green" title="E38">A5</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A''</font>
 <b>and </b><br/><a NAME="E8:44"/><i><font color="Green" title="E39">A6</font></i>: 
<font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">C'</font>
 <b>and </b><br/><a NAME="E9:44"/><i><font color="Green" title="E40">A7</font></i>: 
 <a href="jordan1k.html#K1" title="JORDAN1K:func.1">dist_min</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c7">A''</font>,<font color="Maroon" title="c8">C'</font>
 <b>by </b><i><a class="ref" href="jordan1k.html#D1" target="_self" title="JORDAN1K:def.1">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">f</font> =  <a href="weierstr.html#K6" title="WEIERSTR:func.6">dist_min</a> <font color="Maroon" title="c5">A'</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E11:44"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">X</font> = <font color="Maroon" title="c9">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">C'</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="txt" href="#E8:44"><i><font color="Green" title="E39">A6</font></i></a>, <a class="ref" href="relat_1.html#T152" title="RELAT_1:th.152">RELAT_1:152</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">Y</font> = <font color="Maroon" title="c9">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">D'</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<i><font color="Green" title="E41">A8</font></i>:  <a href="pscomp_1.html#NK3" title="PSCOMP_1:NK.3">inf</a> <font color="Maroon" title="c11">Y</font> = 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="weierstr.html#K1" title="WEIERSTR:func.1">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist_min</a> <font color="Maroon" title="c5">A'</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">D'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="weierstr.html#D3" title="WEIERSTR:def.3">WEIERSTR:def 3</a></i>
<br/>.= 
 <a href="weierstr.html#K3" title="WEIERSTR:func.3">lower_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist_min</a> <font color="Maroon" title="c5">A'</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">D'</font></span>)</span>
<b>by </b><i><a class="ref" href="weierstr.html#D5" title="WEIERSTR:def.5">WEIERSTR:def 5</a></i>
<br/>.= 
 <a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c5">A'</font>,<font color="Maroon" title="c6">D'</font>
<b>by </b><i><a class="ref" href="weierstr.html#D9" title="WEIERSTR:def.9">WEIERSTR:def 9</a></i>
;<br/>
<i><font color="Green" title="E42">A9</font></i>:  <a href="pscomp_1.html#NK3" title="PSCOMP_1:NK.3">inf</a> <font color="Maroon" title="c10">X</font> = 
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="weierstr.html#K1" title="WEIERSTR:func.1">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist_min</a> <font color="Maroon" title="c5">A'</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">C'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="weierstr.html#D3" title="WEIERSTR:def.3">WEIERSTR:def 3</a></i>
<br/>.= 
 <a href="weierstr.html#K3" title="WEIERSTR:func.3">lower_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist_min</a> <font color="Maroon" title="c5">A'</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">C'</font></span>)</span>
<b>by </b><i><a class="ref" href="weierstr.html#D5" title="WEIERSTR:def.5">WEIERSTR:def 5</a></i>
<br/>.= 
 <a href="weierstr.html#K7" title="WEIERSTR:func.7">min_dist_min</a> <font color="Maroon" title="c5">A'</font>,<font color="Maroon" title="c8">C'</font>
<b>by </b><i><a class="ref" href="weierstr.html#D9" title="WEIERSTR:def.9">WEIERSTR:def 9</a></i>
;<br/>
<a NAME="E16:44"/>
<font color="Maroon" title="c11">Y</font> is <a href="xxreal_2.html#V3" title="XXREAL_2:attr.3">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>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_4.html#D2" title="SEQ_4:def.2">SEQ_4:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y</font> or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/>

<b>let </b><font color="Maroon" title="c12">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c12">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y</font> or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">r</font> )</span><br/>

<b>assume </b><a NAME="E1:44_3"/>
<font color="Maroon" title="c12">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">r</font></span><br/>

<a NAME="E2:44_3"/><b>then </b>
 ex <font color="Olive" title="b1">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">D'</font> &amp; <font color="Maroon" title="c12">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<b>hence </b><a NAME="E3:44_3"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">r</font>
 <b>by </b><i><a class="txt" href="#E3:44"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="jordan1k.html#T9" target="_self" title="JORDAN1K:th.9">Th9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E17:44"/>
 <a href="jordan1k.html#K1" title="JORDAN1K:func.1">dist_min</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">D</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jordan1k.html#K1" title="JORDAN1K:func.1">dist_min</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">C</font>
 <b>by </b><i><a class="txt" href="#E1:44"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="#E3:44"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="#E4:44"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="#E5:44"><i><font color="Green" title="E37">A4</font></i></a>, <a class="txt" href="#E7:44"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="#E8:44"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="#E9:44"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="#E14:44"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="#E15:44"><i><font color="Green" title="E42">A9</font></i></a>, <a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a>, <a class="ref" href="seq_4.html#T64" title="SEQ_4:th.64">SEQ_4:64</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
