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

<span class="kw">set </span><font color="Maroon" title="c1">T</font> =  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1;<br/>
<a NAME="E1:29"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
 for <font color="Olive" title="b1">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#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span><br/>  for <font color="Olive" title="b2">U</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</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> 1</span>)</span>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">U</font> holds <br/> ex <font color="Olive" title="b3">W</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</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> 1</span>)</span> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">W</font> &amp; <font color="Olive" title="b3">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">U</font> &amp;  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b3">W</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b3">W</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">p</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#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span>; <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">U</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</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> 1</span>)</span>  st <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> holds <br/> ex <font color="Olive" title="b2">W</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</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> 1</span>)</span> st <br/>( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">W</font> &amp; <font color="Olive" title="b2">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">U</font> &amp;  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b2">W</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b2">W</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1 )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">U</font> be   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</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> 1</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="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">U</font> implies  ex <font color="Olive" title="b1">W</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</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> 1</span>)</span> st <br/>( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">U</font> &amp;  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b1">W</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b1">W</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1 ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:29_1"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">U</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">W</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</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> 1</span>)</span> st <br/>( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">U</font> &amp;  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b1">W</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Olive" title="b1">W</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1 )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">p9</font> = <font color="Maroon" title="c2">p</font> as   <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> 1</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="topdim_2.html#E25"><span class="lab"><font color="Green" title="E17">Lm6</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">M</font> =  <a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 1;<br/>
<a NAME="E3:29_1"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
 <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 <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> 1</span>)</span></span>)</span>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</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> 1</span>)</span></span>)</span> #) <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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</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="euclid.html#D8" title="EUCLID:def.8">EUCLID:def 8</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">V</font> = <font color="Maroon" title="c3">U</font> as   <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> 1</span>)</span></span>)</span> ;<br/>
<a NAME="E5:29_1"/>
<font color="Maroon" title="c6">V</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_1"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="pre_topc.html#T30" title="PRE_TOPC:th.30">PRE_TOPC:30</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">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="E7:29_1"/><span class="lab"><font color="Green" title="E25">A4</font></span>: 
<font color="Maroon" title="c7">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="E8:29_1"/><span class="lab"><font color="Green" title="E26">A5</font></span>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c4">p9</font>,<font color="Maroon" title="c7">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">U</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="topmetr.html#T15" title="TOPMETR:th.15">TOPMETR:15</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">B</font> =  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c4">p9</font>,<font color="Maroon" title="c7">r</font>) as   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</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> 1</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="kurato_2.html#T1" title="KURATO_2:th.1">KURATO_2:1</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">B</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" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c8">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">U</font> &amp;  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c8">B</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1 )</span><br/>

<a NAME="E10:29_1"/>
<font color="Maroon" title="c4">p9</font> is   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of 1, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T131" title="FINSEQ_2:th.131">FINSEQ_2:131</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">p1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E12:29_1"/><span class="lab"><font color="Green" title="E27">A6</font></span>: 
<font color="Maroon" title="c4">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c9">p1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T97" title="FINSEQ_2:th.97">FINSEQ_2:97</a></span>;<br/>
<a NAME="E13:29_1"/><span class="lab"><font color="Green" title="E28">A7</font></span>: 
 <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">r</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c4">p9</font>,<font color="Maroon" title="c7">r</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal9.html#T13" title="TOPREAL9:th.13">TOPREAL9:13</a></span>;<br/>
<a NAME="E14:29_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A8</font></span>: 
 <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c8">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">p1</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c7">r</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>,<span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">p1</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c7">r</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29_1"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E12:29_1"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="ref" href="topdim_2.html#T18" target="_self" title="TOPDIM_2:th.18">Th18</a></span>;<br/>
<a NAME="E15:29_1"/>
 <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1 is  <a href="pcomps_1.html#V3" title="PCOMPS_1:attr.3">metrizable</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:29_1_1"/>
 ex <font color="Olive" title="b1">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> st <br/>( <font color="Olive" title="b1">f</font> <a href="pcomps_1.html#R1" title="PCOMPS_1:pred.1">is_metric_of</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span> &amp;  <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">Family_open_set</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">SpaceMetr</a> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span>,<font color="Olive" title="b1">f</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</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="txt" href="#E3:29_1"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="pcomps_1.html#D8" title="PCOMPS_1:def.8">PCOMPS_1:def 8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:29_1_1"/>
 <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1 is  <a href="pcomps_1.html#V3" title="PCOMPS_1:attr.3">metrizable</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pcomps_1.html#D8" title="PCOMPS_1:def.8">PCOMPS_1:def 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>
<a NAME="E16:29_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c8">B</font></span>)</span> is  <a href="pcomps_1.html#V3" title="PCOMPS_1:attr.3">metrizable</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E17:29_1"/>
( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c8">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">U</font> &amp;  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c8">B</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c8">B</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29_1"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E8:29_1"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E13:29_1"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="topdim_2.html#T19" target="_self" title="TOPDIM_2:th.19">Th19</a>, <a class="txt" href="#E14:29_1"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="ref" href="jordan.html#T16" title="JORDAN:th.16">JORDAN:16</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>
<a NAME="E2:29"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A9</font></span>: 
 <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1 is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topdim_1.html#T15" title="TOPDIM_1:th.15">TOPDIM_1:15</a></span>;<br/>
<a NAME="E3:29"/><span class="lab"><font color="Green" title="E24">A10</font></span>: 
 <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c2">p</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>, <font color="Maroon" title="c3">q</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> as   <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> 1</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T98" title="FINSEQ_2:th.98">FINSEQ_2:98</a></span>;<br/>
<a NAME="E2:29_2"/>
 <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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#D8" title="EUCLID:def.8">EUCLID:def 8</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">p9</font> = <font color="Maroon" title="c2">p</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span> ;<br/>
<a NAME="E4:29_2"/><span class="lab"><font color="Green" title="E25">A11</font></span>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c2">p</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c4">p9</font>,1)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal9.html#T13" title="TOPREAL9:th.13">TOPREAL9:13</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">B</font> =  <a href="topreal9.html#K1" title="TOPREAL9:func.1">Ball</a> (<font color="Maroon" title="c4">p9</font>,1) as   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</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> 1</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="kurato_2.html#T1" title="KURATO_2:th.1">KURATO_2:1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:29_2"/>
 <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E7:29_2"/><span class="kw">then </span>
 <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
;<br/>
<a NAME="E8:29_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A12</font></span>: 
 <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T7" title="INT_1:th.7">INT_1:7</a></span>;<br/>
<a NAME="E9:29_2"/>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan.html#T16" title="JORDAN:th.16">JORDAN:16</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">W</font> being   <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</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> 1</span>)</span><span class="kw"> such that </span><br/><a NAME="E11:29_2"/><span class="lab"><font color="Green" title="E27">A13</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W</font>
 <span class="kw">and </span><br/><a NAME="E12:29_2"/><span class="lab"><font color="Green" title="E28">A14</font></span>: 
<font color="Maroon" title="c6">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font>
 <span class="kw">and </span><br/><a NAME="E13:29_2"/><span class="lab"><font color="Green" title="E29">A15</font></span>: 
(  <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c6">W</font> is  <a href="topdim_1.html#V1" title="TOPDIM_1:attr.1">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K2" title="TOPDIM_1:func.2">ind</a> <span class="p1">(<span class="default"><a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c6">W</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="txt" href="#E8:29_2"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="ref" href="topdim_1.html#T16" title="TOPDIM_1:th.16">TOPDIM_1:16</a></span>;<br/>
<a NAME="E14:29_2"/>
 <a href="tops_1.html#K2" title="TOPS_1:func.2">Fr</a> <font color="Maroon" title="c6">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <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="txt" href="#E13:29_2"><span class="lab"><font color="Green" title="E29">A15</font></span></a></span>;<br/>
<a NAME="E15:29_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A16</font></span>: 
<font color="Maroon" title="c6">W</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="ref" href="topgen_1.html#T14" title="TOPGEN_1:th.14">TOPGEN_1:14</a></span>;<br/>
<a NAME="E16:29_2"/><span class="lab"><font color="Green" title="E31">A17</font></span>: 
<font color="Maroon" title="c6">W</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">W</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span> <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> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T2" title="PRE_TOPC:th.2">PRE_TOPC:2</a></span>;<br/>
<a NAME="E17:29_2"/>
<font color="Maroon" title="c6">W</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">W</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T24" title="SUBSET_1:th.24">SUBSET_1:24</a></span>;<br/>
<a NAME="E18:29_2"/><span class="kw">then </span>
(  <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> 1</span>)</span> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  &amp; <font color="Maroon" title="c6">W</font>,<font color="Maroon" title="c6">W</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:29_2"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="ref" href="connsp_1.html#T2" title="CONNSP_1:th.2">CONNSP_1:2</a>, <a class="ref" href="jordan2c.html#T18" title="JORDAN2C:th.18">JORDAN2C:18</a></span>;<br/>
<a NAME="E19:29_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A18</font></span>: 
<font color="Maroon" title="c6">W</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <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="txt" href="#E11:29_2"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="txt" href="#E16:29_2"><span class="lab"><font color="Green" title="E31">A17</font></span></a>, <a class="ref" href="connsp_1.html#T15" title="CONNSP_1:th.15">CONNSP_1:15</a></span>;<br/>
<a NAME="E20:29_2"/><span class="lab"><font color="Green" title="E33">A19</font></span>: 
<font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">s</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> where <font color="Olive" title="b1">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_2"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="ref" href="jordan2b.html#T27" title="JORDAN2B:th.27">JORDAN2B:27</a></span>;<br/>
<a NAME="E21:29_2"/><span class="lab"><font color="Green" title="E34">A20</font></span>: 
 <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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#D8" title="EUCLID:def.8">EUCLID:def 8</a></span>;<br/>
<a NAME="E22:29_2"/>
 not <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:29_2_1"/>
<font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:29_2_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">s</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> &amp; <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</font> &amp; <font color="Olive" title="b1">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:29_2"><span class="lab"><font color="Green" title="E33">A19</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:29_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T76" title="FINSEQ_1:th.76">FINSEQ_1:76</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>
<a NAME="E23:29_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:29_2"><span class="lab"><font color="Green" title="E28">A14</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E24:29_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:29_2"><span class="lab"><font color="Green" title="E34">A20</font></span></a>, <a class="txt" href="#E19:29_2"><span class="lab"><font color="Green" title="E32">A18</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</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>
<a NAME="E4:29"/>
 <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="ref" href="topdim_1.html#T16" title="TOPDIM_1:th.16">TOPDIM_1:16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:29"/>
(  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1 is  <a href="topdim_1.html#V3" title="TOPDIM_1:attr.3">with_finite_small_inductive_dimension</a>  &amp;  <a href="topdim_1.html#K4" title="TOPDIM_1:func.4">ind</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E3:29"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="ref" href="topdim_1.html#T15" title="TOPDIM_1:th.15">TOPDIM_1:15</a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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>
