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<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">P</font>, <font color="Olive" title="b2">Q</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#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>  st <font color="Olive" title="b1">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">P</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  &amp; <font color="Olive" title="b2">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b2">Q</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  holds <br/> ex <font color="Olive" title="b3">x1</font>, <font color="Olive" title="b4">x2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> st <br/>( <font color="Olive" title="b3">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <font color="Olive" title="b4">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">Q</font> &amp;  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b3">x1</font>,<font color="Olive" title="b4">x2</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="Olive" title="b1">P</font>,<font color="Olive" title="b2">Q</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">P</font>, <font color="Maroon" title="c3">Q</font> be   <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#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c2">P</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  &amp; <font color="Maroon" title="c3">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">Q</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  implies  ex <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> st <br/>( <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> &amp;  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</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="c2">P</font>,<font color="Maroon" title="c3">Q</font> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:34"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
( <font color="Maroon" title="c2">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c2">P</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  )
 <span class="kw">and </span><br/><a NAME="E2:34"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
( <font color="Maroon" title="c3">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">Q</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> st <br/>( <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> &amp;  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b1">x1</font>,<font color="Olive" title="b2">x2</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="c2">P</font>,<font color="Maroon" title="c3">Q</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:34"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<font color="Maroon" title="c4">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font>
 <span class="kw">and </span><br/><a NAME="E5:34"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
<span class="p1">(<span class="default"><a href="weierstr.html#K6" title="WEIERSTR:func.6">dist_min</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c2">P</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">Q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E2:34"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="weierstr.html#T21" target="_self" title="WEIERSTR:th.21">Th21</a>, <a class="ref" href="weierstr.html#T33" target="_self" title="WEIERSTR:th.33">Th33</a></span>;<br/>
<a NAME="E6:34"/><span class="lab"><font color="Green" title="E27">A5</font></span>: 
 <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">Family_open_set</a> <font color="Maroon" title="c1">M</font></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pcomps_1.html#D6" title="PCOMPS_1:def.6">PCOMPS_1:def 6</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">x2</font> = <font color="Maroon" title="c4">x2</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">M</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E9:34"/><span class="lab"><font color="Green" title="E28">A6</font></span>: 
<font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font>
 <span class="kw">and </span><br/><a NAME="E10:34"/><span class="lab"><font color="Green" title="E29">A7</font></span>: 
<span class="p1">(<span class="default"><a href="weierstr.html#K4" title="WEIERSTR:func.4">dist</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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#K4" title="WEIERSTR:func.4">dist</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c2">P</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="weierstr.html#T21" target="_self" title="WEIERSTR:th.21">Th21</a>, <a class="ref" href="weierstr.html#T22" target="_self" title="WEIERSTR:th.22">Th22</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">x1</font> = <font color="Maroon" title="c6">x1</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:34"><span class="lab"><font color="Green" title="E27">A5</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">x1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> st <br/>( <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Olive" title="b1">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> &amp;  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c7">x1</font>,<font color="Olive" title="b1">x2</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="c2">P</font>,<font color="Maroon" title="c3">Q</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c5">x2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> &amp;  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c5">x2</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="c2">P</font>,<font color="Maroon" title="c3">Q</font> )</span><br/>

 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c5">x2</font> = 
<span class="p1">(<span class="default"><a href="weierstr.html#K4" title="WEIERSTR:func.4">dist</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="weierstr.html#D6" target="_self" title="WEIERSTR:def.6">Def6</a></span>
<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="c2">P</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">Q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="txt" href="#E10:34"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="ref" href="weierstr.html#D8" target="_self" title="WEIERSTR:def.8">Def8</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E13:34"/>
( <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> &amp;  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c5">x2</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="c2">P</font>,<font color="Maroon" title="c3">Q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E9:34"><span class="lab"><font color="Green" title="E28">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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