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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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">V</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> <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 <font color="Olive" title="b1">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  holds <br/>for <font color="Olive" title="b2">e</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <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>  st <font color="Olive" title="b2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> holds <br/> ex <font color="Olive" title="b3">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  <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> <font color="Olive" title="b2">e</font>,<span class="p1">(<span class="default">1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">V</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</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="c2">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies for <font color="Olive" title="b1">e</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <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>  st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> holds <br/> ex <font color="Olive" title="b2">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  <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> <font color="Olive" title="b1">e</font>,<span class="p1">(<span class="default">1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:49"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
<font color="Maroon" title="c2">V</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"> for <font color="Olive" title="b1">e</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <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>  st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> holds <br/> ex <font color="Olive" title="b2">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  <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> <font color="Olive" title="b1">e</font>,<span class="p1">(<span class="default">1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">e</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <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>; <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="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> implies  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  <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> <font color="Maroon" title="c3">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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:49"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</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">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  <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> <font color="Maroon" title="c3">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> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">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="E4:49"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
<font color="Maroon" title="c4">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="E5:49"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c3">e</font>,<font color="Maroon" title="c4">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E21">A1</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="c5">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="E7:49"/><span class="lab"><font color="Green" title="E24">A4</font></span>: 
 <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> <font color="Maroon" title="c3">e</font>,<span class="p1">(<span class="default">1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c5">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="c3">e</font>,<font color="Maroon" title="c4">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:49"><span class="lab"><font color="Green" title="E22">A2</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="c5">m</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#K4" title="EUCLID_9:func.4">OpenHypercube</a> <font color="Maroon" title="c3">e</font>,<span class="p1">(<span class="default">1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font></span><br/>

<span class="kw">thus </span><a NAME="E8:49"/>
 <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> <font color="Maroon" title="c3">e</font>,<span class="p1">(<span class="default">1 <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:49"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="txt" href="#E7:49"><span class="lab"><font color="Green" title="E24">A4</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/>


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