<?xml version="1.0"?>
<div class="add">

<span class="kw">set </span><font color="Maroon" title="c3">T</font> =  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>
<a NAME="E1:52_1_1"/>
 <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> <font color="Maroon" title="c2">e</font>,<span class="p1">(<span class="default">1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_9.html#K5" title="EUCLID_9:func.5">OpenHypercubes</a> <font color="Maroon" title="c2">e</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E2:52_1_1"/>
not  <a href="euclid_9.html#K5" title="EUCLID_9:func.5">OpenHypercubes</a> <font color="Maroon" title="c2">e</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="euclid_9.html#K5" title="EUCLID_9:func.5">OpenHypercubes</a> <font color="Maroon" title="c2">e</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  &amp;  <a href="euclid_9.html#K5" title="EUCLID_9:func.5">OpenHypercubes</a> <font color="Maroon" title="c2">e</font> is  <a href="euclid_9.html#K2" title="EUCLID_9:func.2">@</a> <font color="Maroon" title="c2">e</font> <a href="yellow_8.html#V1" title="YELLOW_8:attr.1">-quasi_basis</a>  )</span><br/>

<div><span class="kw">hereby </span> <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="euclid_9.html#K5" title="EUCLID_9:func.5">OpenHypercubes</a> <font color="Maroon" title="c2">e</font> is  <a href="euclid_9.html#K2" title="EUCLID_9:func.2">@</a> <font color="Maroon" title="c2">e</font> <a href="yellow_8.html#V1" title="YELLOW_8:attr.1">-quasi_basis</a> </span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c4">A</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> <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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_9.html#K5" title="EUCLID_9:func.5">OpenHypercubes</a> <font color="Maroon" title="c2">e</font> implies <font color="Maroon" title="c4">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:52_1_1_1"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_9.html#K5" title="EUCLID_9:func.5">OpenHypercubes</a> <font color="Maroon" title="c2">e</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="c4">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/><a NAME="E2:52_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">m</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>   <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>  st <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> <font color="Maroon" title="c2">e</font>,<span class="p1">(<span class="default">1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Olive" title="b1">m</font></span>)</span>
 ;<br/><span class="kw">hence </span><a NAME="E3:52_1_1_1"/>
<font color="Maroon" title="c4">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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="c4">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_9.html#K5" title="EUCLID_9:func.5">OpenHypercubes</a> <font color="Maroon" title="c2">e</font> implies  <a href="euclid_9.html#K2" title="EUCLID_9:func.2">@</a> <font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> )</span><br/><span class="kw">assume </span><a NAME="E1:52_1_1_2"/>
<font color="Maroon" title="c4">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_9.html#K5" title="EUCLID_9:func.5">OpenHypercubes</a> <font color="Maroon" title="c2">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="euclid_9.html#K2" title="EUCLID_9:func.2">@</a> <font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font></span><br/><a NAME="E2:52_1_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">m</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>   <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>  st <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> <font color="Maroon" title="c2">e</font>,<span class="p1">(<span class="default">1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Olive" title="b1">m</font></span>)</span>
 ;<br/><span class="kw">hence </span><a NAME="E3:52_1_1_2"/>
 <a href="euclid_9.html#K2" title="EUCLID_9:func.2">@</a> <font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_9.html#T11" target="_self" title="EUCLID_9:th.11">Th11</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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:52_1_1"/>
 <a href="euclid_9.html#K2" title="EUCLID_9:func.2">@</a> <font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <span class="p1">(<span class="default"><a href="euclid_9.html#K5" title="EUCLID_9:func.5">OpenHypercubes</a> <font color="Maroon" title="c2">e</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="yellow_8.html#D2" title="YELLOW_8:def.2">YELLOW_8:def 2</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">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K21</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#K3" title="PCOMPS_1:func.3">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>)  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  or  not  <a href="euclid_9.html#K2" title="EUCLID_9:func.2">@</a> <font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive">b<sub>1</sub></font> or  ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K21</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#K3" title="PCOMPS_1:func.3">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">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_9.html#K5" title="EUCLID_9:func.5">OpenHypercubes</a> <font color="Maroon" title="c2">e</font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive">b<sub>1</sub></font> ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">S</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> <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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">S</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  or  not  <a href="euclid_9.html#K2" title="EUCLID_9:func.2">@</a> <font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font> or  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K21</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#K3" title="PCOMPS_1:func.3">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">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_9.html#K5" title="EUCLID_9:func.5">OpenHypercubes</a> <font color="Maroon" title="c2">e</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">S</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:52_1_1"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
<font color="Maroon" title="c4">S</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E7:52_1_1"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
 <a href="euclid_9.html#K2" title="EUCLID_9:func.2">@</a> <font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</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">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K21</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#K3" title="PCOMPS_1:func.3">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">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_9.html#K5" title="EUCLID_9:func.5">OpenHypercubes</a> <font color="Maroon" title="c2">e</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">S</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">r</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E9:52_1_1"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
<font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E10:52_1_1"/><span class="lab"><font color="Green" title="E24">A4</font></span>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c5">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:52_1_1"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E7:52_1_1"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="topmetr.html#T22" title="TOPMETR:th.22">TOPMETR:22</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">m</font> being  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>   <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> <span class="kw"> such that </span><br/><a NAME="E12:52_1_1"/><span class="lab"><font color="Green" title="E25">A5</font></span>: 
 <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> <font color="Maroon" title="c2">e</font>,<span class="p1">(<span class="default">1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c5">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:52_1_1"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="ref" href="goboard6.html#T4" title="GOBOARD6:th.4">GOBOARD6:4</a>, <a class="ref" href="euclid_9.html#T19" target="_self" title="EUCLID_9:th.19">Th19</a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c7">V</font> =  <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> <font color="Maroon" title="c2">e</font>,<span class="p1">(<span class="default">1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c6">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="c7">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_9.html#K5" title="EUCLID_9:func.5">OpenHypercubes</a> <font color="Maroon" title="c2">e</font> &amp; <font color="Maroon" title="c7">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">S</font> )</span><br/>

<span class="kw">thus </span><a NAME="E13:52_1_1"/>
<font color="Maroon" title="c7">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_9.html#K5" title="EUCLID_9:func.5">OpenHypercubes</a> <font color="Maroon" title="c2">e</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">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">S</font></span><br/>

<span class="kw">thus </span><a NAME="E14:52_1_1"/>
<font color="Maroon" title="c7">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:52_1_1"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="txt" href="#E12:52_1_1"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</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/>


</div>
