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<div class="add">

<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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</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>  st <font color="Olive" title="b1">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  &amp; <font color="Olive" title="b1">A</font> is  <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>  holds <br/><font color="Olive" title="b1">A</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> </span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">A</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> $1</span>)</span>  st <font color="Olive" title="b1">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  &amp; <font color="Olive" title="b1">A</font> is  <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>  holds <br/><font color="Olive" title="b1">A</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> ;<br/>
<a NAME="E1:38"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:38_1"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:38_1_1"/>
( <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or <font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:38_1_1_1"/>
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:38_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="simplex2.html#E32"><span class="lab"><font color="Green" title="E24">Lm4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:38_1_1_2"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1]</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c3">n1</font> = <font color="Maroon" title="c2">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1;<br/><span class="kw">set </span><font color="Maroon" title="c4">TR1</font> =  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1;<br/><span class="kw">set </span><font color="Maroon" title="c5">TRn</font> =  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font>;<br/><span class="kw">set </span><font color="Maroon" title="c6">TRn1</font> =  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>;<br/><span class="kw">let </span><font color="Maroon" title="c7">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="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  &amp; <font color="Maroon" title="c7">A</font> is  <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>  implies <font color="Maroon" title="c7">A</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:38_1_1_2"/><span class="lab"><font color="Green" title="E29">A4</font></span>: 
<font color="Maroon" title="c7">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">and </span><br/><a NAME="E3:38_1_1_2"/><span class="lab"><font color="Green" title="E30">A5</font></span>: 
<font color="Maroon" title="c7">A</font> is  <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">A</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> </span><br/><span class="kw">consider </span><font color="Maroon" title="c8">p</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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span></span>)</span>, <font color="Maroon" title="c9">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="E5:38_1_1_2"/><span class="lab"><font color="Green" title="E31">A6</font></span>: 
<font color="Maroon" title="c7">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">r</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1_1_2"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="ref" href="simplex2.html#T21" target="_self" title="SIMPLEX2:th.21">Th21</a></span>;<br/><a NAME="E6:38_1_1_2"/><span class="lab"><font color="Green" title="E32">A7</font></span>: 
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span>,<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E8:38_1_1_2"/><span class="lab"><font color="Green" title="E33">A8</font></span>: 
<font color="Maroon" title="c10">f</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 <span class="kw">and </span><br/><a NAME="E9:38_1_1_2"/><span class="lab"><font color="Green" title="E34">A9</font></span>: 
 for <font color="Olive" title="b1">fi</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span><br/>  for <font color="Olive" title="b2">fj</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span> holds  <font color="Maroon" title="c10">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b1">fi</font>,<font color="Olive" title="b2">fj</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">fi</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b2">fj</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex2.html#T19" target="_self" title="SIMPLEX2:th.19">Th19</a></span>;<br/><a NAME="E10:38_1_1_2"/><span class="lab"><font color="Green" title="E35">A10</font></span>: 
( <font color="Maroon" title="c10">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:38_1_1_2"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="ref" href="topgrp_1.html#T24" title="TOPGRP_1:th.24">TOPGRP_1:24</a></span>;<br/><a NAME="E11:38_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">q1</font>, <font color="Maroon" title="c12">q2</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E13:38_1_1_2"/><span class="lab"><font color="Green" title="E36">A11</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">and </span><br/><a NAME="E14:38_1_1_2"/><span class="lab"><font color="Green" title="E37">A12</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">and </span><br/><a NAME="E15:38_1_1_2"/><span class="lab"><font color="Green" title="E38">A13</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">q1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c12">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T22" title="FINSEQ_2:th.22">FINSEQ_2:22</a></span>;<br/><a NAME="E16:38_1_1_2"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c8">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ;<br/><a NAME="E17:38_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A14</font></span>: 
<font color="Maroon" title="c8">p</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/><a NAME="E18:38_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A15</font></span>: 
<font color="Maroon" title="c11">q1</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:38_1_1_2"><span class="lab"><font color="Green" title="E38">A13</font></span></a>, <a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></span>;<br/><a NAME="E19:38_1_1_2"/><span class="lab"><font color="Green" title="E41">A16</font></span>: 
<font color="Maroon" title="c12">q2</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:38_1_1_2"><span class="lab"><font color="Green" title="E38">A13</font></span></a>, <a class="txt" href="#E17:38_1_1_2"><span class="lab"><font color="Green" title="E39">A14</font></span></a>, <a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">q1</font> = <font color="Maroon" title="c11">q1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c2">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:38_1_1_2"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="txt" href="#E18:38_1_1_2"><span class="lab"><font color="Green" title="E40">A15</font></span></a>, <a class="ref" href="topreal7.html#T16" title="TOPREAL7:th.16">TOPREAL7:16</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">q2</font> = <font color="Maroon" title="c12">q2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 1</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:38_1_1_2"><span class="lab"><font color="Green" title="E37">A12</font></span></a>, <a class="txt" href="#E19:38_1_1_2"><span class="lab"><font color="Green" title="E41">A16</font></span></a>, <a class="ref" href="topreal7.html#T16" title="TOPREAL7:th.16">TOPREAL7:16</a></span>;<br/><a NAME="E22:38_1_1_2"/><span class="lab"><font color="Green" title="E42">A17</font></span>: 
<font color="Maroon" title="c10">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:38_1_1_2"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">Bn</font> =  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c13">q1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span>) as   <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="c2">n</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal3.html#T8" title="TOPREAL3:th.8">TOPREAL3:8</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">B1</font> =  <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c14">q2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 1</span>)</span></span>)</span>) as   <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> 1</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal3.html#T8" title="TOPREAL3:th.8">TOPREAL3:8</a></span>;<br/><a NAME="E25:38_1_1_2"/>
(  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c14">q2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 1</span>)</span></span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">B1</font> &amp;  <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Maroon" title="c14">q2</font>,<font color="Maroon" title="c9">r</font>) <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="c14">q2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 1</span>)</span></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_9.html#T18" title="EUCLID_9:th.18">EUCLID_9:18</a>, <a class="ref" href="metric_1.html#T14" title="METRIC_1:th.14">METRIC_1:14</a></span>;<br/><a NAME="E26:38_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A18</font></span>: 
 <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Maroon" title="c14">q2</font>,<font color="Maroon" title="c9">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">B1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E27:38_1_1_2"/>
(  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c13">q1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">Bn</font> &amp;  <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Maroon" title="c13">q1</font>,<font color="Maroon" title="c9">r</font>) <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="c13">q1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1_1_2"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="euclid_9.html#T18" title="EUCLID_9:th.18">EUCLID_9:18</a>, <a class="ref" href="metric_1.html#T14" title="METRIC_1:th.14">METRIC_1:14</a></span>;<br/><a NAME="E28:38_1_1_2"/><span class="kw">then </span>
 <a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Maroon" title="c13">q1</font>,<font color="Maroon" title="c9">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">Bn</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E29:38_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A19</font></span>: 
<span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Maroon" title="c13">q1</font>,<font color="Maroon" title="c9">r</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Maroon" title="c14">q2</font>,<font color="Maroon" title="c9">r</font>)</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c15">Bn</font>,<font color="Maroon" title="c16">B1</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:38_1_1_2"><span class="lab"><font color="Green" title="E43">A18</font></span></a>, <a class="ref" href="zfmisc_1.html#T96" title="ZFMISC_1:th.96">ZFMISC_1:96</a></span>;<br/><a NAME="E30:38_1_1_2"/>
 <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c14">q2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 1</span>)</span></span>)</span>) is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal6.html#T59" title="TOPREAL6:th.59">TOPREAL6:59</a></span>;<br/><a NAME="E31:38_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">B1</font> is  <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#D1" title="JORDAN2C:def.1">JORDAN2C:def 1</a></span>;<br/><a NAME="E32:38_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A20</font></span>: 
<font color="Maroon" title="c16">B1</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="simplex2.html#E32"><span class="lab"><font color="Green" title="E24">Lm4</font></span></a>, <a class="ref" href="topreal6.html#T58" title="TOPREAL6:th.58">TOPREAL6:58</a></span>;<br/><a NAME="E33:38_1_1_2"/>
 <a href="metric_1.html#K10" title="METRIC_1:func.10">cl_Ball</a> (<font color="Maroon" title="c13">q1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K6" title="SQUARE_1:func.6">sqrt</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span>) is  <a href="tbsp_1.html#V6" title="TBSP_1:attr.6">bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal6.html#T59" title="TOPREAL6:th.59">TOPREAL6:59</a></span>;<br/><a NAME="E34:38_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">Bn</font> is  <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#D1" title="JORDAN2C:def.1">JORDAN2C:def 1</a></span>;<br/><a NAME="E35:38_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">Bn</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E6:38_1_1_2"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="ref" href="topreal6.html#T58" title="TOPREAL6:th.58">TOPREAL6:58</a></span>;<br/><a NAME="E36:38_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A21</font></span>: 
<span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><font color="Maroon" title="c15">Bn</font>,<font color="Maroon" title="c16">B1</font></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> is   <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span></span><a href="borsuk_1.html#K2" title="BORSUK_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:38_1_1_2"><span class="lab"><font color="Green" title="E45">A20</font></span></a>, <a class="ref" href="borsuk_3.html#T23" title="BORSUK_3:th.23">BORSUK_3:23</a></span>;<br/><a NAME="E37:38_1_1_2"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:38_1_1_2"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="ref" href="topgrp_1.html#T24" title="TOPGRP_1:th.24">TOPGRP_1:24</a></span>;<br/><a NAME="E38:38_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A22</font></span>: 
<font color="Maroon" title="c10">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T77" title="FUNCT_1:th.77">FUNCT_1:77</a></span>;<br/><a NAME="E39:38_1_1_2"/>
<font color="Maroon" title="c10">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Maroon" title="c13">q1</font>,<font color="Maroon" title="c9">r</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Maroon" title="c14">q2</font>,<font color="Maroon" title="c9">r</font>)</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span> <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="c8">p</font>,<font color="Maroon" title="c9">r</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:38_1_1_2"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="txt" href="#E13:38_1_1_2"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="txt" href="#E15:38_1_1_2"><span class="lab"><font color="Green" title="E38">A13</font></span></a>, <a class="ref" href="simplex2.html#T20" target="_self" title="SIMPLEX2:th.20">Th20</a></span>;<br/><a NAME="E40:38_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A23</font></span>: 
<font color="Maroon" title="c10">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c7">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Maroon" title="c13">q1</font>,<font color="Maroon" title="c9">r</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="euclid_9.html#K4" title="EUCLID_9:func.4">OpenHypercube</a> (<font color="Maroon" title="c14">q2</font>,<font color="Maroon" title="c9">r</font>)</span>)</span></span><a href="borsuk_1.html#K3" title="BORSUK_1:func.3">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:38_1_1_2"><span class="lab"><font color="Green" title="E31">A6</font></span></a>, <a class="txt" href="#E10:38_1_1_2"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="txt" href="#E38:38_1_1_2"><span class="lab"><font color="Green" title="E47">A22</font></span></a>, <a class="ref" href="funct_1.html#T87" title="FUNCT_1:th.87">FUNCT_1:87</a></span>;<br/><a NAME="E41:38_1_1_2"/>
<font color="Maroon" title="c10">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c7">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38_1_1_2"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="txt" href="#E8:38_1_1_2"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="ref" href="topgrp_1.html#T24" title="TOPGRP_1:th.24">TOPGRP_1:24</a></span>;<br/><a NAME="E42:38_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c7">A</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:38_1_1_2"><span class="lab"><font color="Green" title="E44">A19</font></span></a>, <a class="txt" href="#E36:38_1_1_2"><span class="lab"><font color="Green" title="E46">A21</font></span></a>, <a class="txt" href="#E40:38_1_1_2"><span class="lab"><font color="Green" title="E48">A23</font></span></a>, <a class="ref" href="compts_1.html#T9" title="COMPTS_1:th.9">COMPTS_1:9</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">hence </span><a NAME="E43:38_1_1_2"/>
<font color="Maroon" title="c7">A</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:38_1_1_2"><span class="lab"><font color="Green" title="E42">A17</font></span></a>, <a class="txt" href="#E38:38_1_1_2"><span class="lab"><font color="Green" title="E47">A22</font></span></a>, <a class="ref" href="weierstr.html#T8" title="WEIERSTR:th.8">WEIERSTR:8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E2:38"/><span class="lab"><font color="Green" title="E27">A24</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ]
 
;<br/>
<a NAME="E3:38"/>
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E27">A24</font></span></a>, <a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E26">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:38"/>
 for <font color="Olive" title="b1">A</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>  st <font color="Olive" title="b1">A</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>  &amp; <font color="Olive" title="b1">A</font> is  <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>  holds <br/><font color="Olive" title="b1">A</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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
