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<div class="add">

<b>let </b><font color="Maroon" title="c1">N</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">G</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span>  st <font color="Olive" title="b1">G</font> is   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">Cover</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span> &amp; <font color="Olive" title="b1">G</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> &amp;  <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> holds <br/> ex <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">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>  &amp; ( for <font color="Olive" title="b3">w1</font>, <font color="Olive" title="b4">w2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font>  st  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b3">w1</font>,<font color="Olive" title="b4">w2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">r</font> holds <br/> ex <font color="Olive" title="b5">Ga</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#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span> st <br/>( <font color="Olive" title="b3">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">Ga</font> &amp; <font color="Olive" title="b4">w2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">Ga</font> &amp; <font color="Olive" title="b5">Ga</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">G</font> ) ) )</span><br/><b>let </b><font color="Maroon" title="c2">G</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">G</font> is   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">Cover</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span> &amp; <font color="Maroon" title="c2">G</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> &amp;  <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> implies  ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">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>  &amp; ( for <font color="Olive" title="b2">w1</font>, <font color="Olive" title="b3">w2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font>  st  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b2">w1</font>,<font color="Olive" title="b3">w2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">r</font> holds <br/> ex <font color="Olive" title="b4">Ga</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#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span> st <br/>( <font color="Olive" title="b2">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">Ga</font> &amp; <font color="Olive" title="b3">w2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">Ga</font> &amp; <font color="Olive" title="b4">Ga</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> ) ) ) )</span><br/>



<b>assume </b><a NAME="E1:7"/><i><font color="Green" title="E7">A1</font></i>: 
( <font color="Maroon" title="c2">G</font> is   <a href="struct_0.html#NM11" title="STRUCT_0:NM.11">Cover</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span> &amp; <font color="Maroon" title="c2">G</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> &amp;  <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">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>  &amp; ( for <font color="Olive" title="b2">w1</font>, <font color="Olive" title="b3">w2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font>  st  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b2">w1</font>,<font color="Olive" title="b3">w2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">r</font> holds <br/> ex <font color="Olive" title="b4">Ga</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#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span> st <br/>( <font color="Olive" title="b2">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">Ga</font> &amp; <font color="Olive" title="b3">w2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">Ga</font> &amp; <font color="Olive" title="b4">Ga</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> ) ) )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:7_1"/><i><font color="Green" title="E8">A2</font></i>: 
for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  holds <br/>(  not <font color="Olive" title="b1">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>  or  ex <font color="Olive" title="b2">w1</font>, <font color="Olive" title="b3">w2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font> st <br/>(  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b2">w1</font>,<font color="Olive" title="b3">w2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">r</font> &amp; ( for <font color="Olive" title="b4">Ga</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#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span>  holds <br/>(  not <font color="Olive" title="b2">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">Ga</font> or  not <font color="Olive" title="b3">w2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">Ga</font> or  not <font color="Olive" title="b4">Ga</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> ) ) ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">w1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b2">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 holds <br/> ex <font color="Olive" title="b3">w2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font> st <br/>(  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b2">w1</font>,<font color="Olive" title="b3">w2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; ( for <font color="Olive" title="b4">Ga</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#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span>  holds <br/>(  not <font color="Olive" title="b2">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">Ga</font> or  not <font color="Olive" title="b3">w2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">Ga</font> or  not <font color="Olive" title="b4">Ga</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> ) ) );<br/><a NAME="E2:7_1"/><i><font color="Green" title="E9">A3</font></i>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ex <font color="Olive" title="b2">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">N</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Olive" title="b2">u</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  implies  ex <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">N</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">e</font>,<font color="Olive" title="b1">u</font>] ) )</span><br/>

<b>assume </b><a NAME="E1:7_1_1"/>
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">N</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">e</font>,<font color="Olive" title="b1">u</font>] )</span><br/>

<b>then reconsider </b><font color="Maroon" title="c4">m</font> = <font color="Maroon" title="c3">e</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<b>set </b><font color="Maroon" title="c5">r</font> = 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/>
<a NAME="E3:7_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T124" title="XREAL_1:th.124">XREAL_1:124</a></i>;<br/>
<a NAME="E4:7_1_1"/><b>then </b>
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217" title="XCMPLX_1:th.217">XCMPLX_1:217</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">w1</font>, <font color="Maroon" title="c7">w2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font><b> such that </b><br/><a NAME="E6:7_1_1"/><i><font color="Green" title="E10">A4</font></i>: 
(  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c6">w1</font>,<font color="Maroon" title="c7">w2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; ( for <font color="Olive" title="b1">Ga</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#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span>  holds <br/>(  not <font color="Maroon" title="c6">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Ga</font> or  not <font color="Maroon" title="c7">w2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Ga</font> or  not <font color="Olive" title="b1">Ga</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> ) ) )
 <b>by </b><i><a class="txt" href="uniform1.html#E1:7_1"><i><font color="Green" title="E8">A2</font></i></a></i>;<br/>
<a NAME="E7:7_1_1"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">w4</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font> &amp; <font color="Olive" title="b2">w4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">w1</font> holds <br/> ex <font color="Olive" title="b3">w5</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font> st <br/>(  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b2">w4</font>,<font color="Olive" title="b3">w5</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; ( for <font color="Olive" title="b4">Ga</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#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span>  holds <br/>(  not <font color="Olive" title="b2">w4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">Ga</font> or  not <font color="Olive" title="b3">w5</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">Ga</font> or  not <font color="Olive" title="b4">Ga</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> ) ) )
 
<b>by </b><i><a class="txt" href="uniform1.html#E6:7_1_1"><i><font color="Green" title="E10">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:7_1_1"/>
 ex <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">N</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">e</font>,<font color="Olive" title="b1">u</font>] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E3:7_1"/>
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">N</font> st <br/>for <font color="Olive" title="b2">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">e</font>,<font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">e</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="uniform1.html#E2:7_1"><i><font color="Green" title="E9">A3</font></i></a>);<br/></i><b>then consider </b><font color="Maroon" title="c3">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">N</font><b> such that </b><br/><a NAME="E5:7_1"/><i><font color="Green" title="E10">A5</font></i>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/>for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b3">w1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font>  st <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> &amp; <font color="Olive" title="b3">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> holds <br/> ex <font color="Olive" title="b4">w2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font> st <br/>(  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b3">w1</font>,<font color="Olive" title="b4">w2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; ( for <font color="Olive" title="b5">Ga</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#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span>  holds <br/>(  not <font color="Olive" title="b3">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">Ga</font> or  not <font color="Olive" title="b4">w2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b5">Ga</font> or  not <font color="Olive" title="b5">Ga</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> ) ) )
 ;<br/><b>set </b><font color="Maroon" title="c4">TM</font> =  <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font>;<br/><a NAME="E6:7_1"/><i><font color="Green" title="E11">A6</font></i>: 
 <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">N</font>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">Family_open_set</a> <font color="Maroon" title="c1">N</font></span>)</span> #)
 <b>by </b><i><a class="ref" href="pcomps_1.html#D6" title="PCOMPS_1:def.6">PCOMPS_1:def 6</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <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#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span>] <b>means </b> ex <font color="Olive" title="b1">p</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">x</font> where <font color="Olive" title="b2">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">N</font> :  ex <font color="Olive" title="b3">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">n</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">n</font> ) </span>}</span> ;<br/><b>consider </b><font color="Maroon" title="c5">F</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span><b> such that </b><br/><a NAME="E8:7_1"/><i><font color="Green" title="E12">A7</font></i>: 
for <font color="Olive" title="b1">B</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#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font> iff <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">B</font>] )
 <b>from </b><i><a class="ref" href="subset_1.html#S3" title="SUBSET_1:sch.3">SUBSET_1:sch 3</a>();<br/></i><a NAME="E9:7_1"/><i><font color="Green" title="E13">A8</font></i>: 
for <font color="Olive" title="b1">p</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <span class="p1">{<span class="default"> <font color="Olive" title="b2">x</font> where <font color="Olive" title="b2">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">N</font> :  ex <font color="Olive" title="b3">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">n</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">n</font> ) </span>}</span>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">p</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">N</font> :  ex <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c6">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font> ) </span>}</span>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<a NAME="E1:7_1_2"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">N</font> :  ex <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c6">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font> ) </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E2:7_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">N</font> :  ex <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c6">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font> ) </span>}</span>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> );<br/><b>set </b><font color="Maroon" title="c6">B0</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">N</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span> ;<br/><a NAME="E10:7_1"/><i><font color="Green" title="E14">A9</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">N</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span>  is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font>
 <b>from </b><i><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></i><a NAME="E11:7_1"/><b>then </b><i><font color="Green" title="E15">A10</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">N</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>] </span>}</span>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font>
 <b>by </b><i><a class="txt" href="uniform1.html#E6:7_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="uniform1.html#E8:7_1"><i><font color="Green" title="E12">A7</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">F</font> = <font color="Maroon" title="c5">F</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span> ;<br/><b>set </b><font color="Maroon" title="c8">G1</font> =  <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">clf</a> <font color="Maroon" title="c7">F</font>;<br/><b>reconsider </b><font color="Maroon" title="c9">B0</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">N</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>] </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="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span> <b>by </b><i><a class="txt" href="uniform1.html#E6:7_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="uniform1.html#E10:7_1"><i><font color="Green" title="E14">A9</font></i></a></i>;<br/><a NAME="E14:7_1"/><i><font color="Green" title="E16">A11</font></i>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c9">B0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">clf</a> <font color="Maroon" title="c7">F</font>
 <b>by </b><i><a class="txt" href="uniform1.html#E11:7_1"><i><font color="Green" title="E15">A10</font></i></a>, <a class="ref" href="pcomps_1.html#D3" title="PCOMPS_1:def.3">PCOMPS_1:def 3</a></i>;<br/><a NAME="E15:7_1"/><i><font color="Green" title="E17">A12</font></i>: 
 <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">clf</a> <font color="Maroon" title="c7">F</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a>
 <b>by </b><i><a class="ref" href="pcomps_1.html#T14" title="PCOMPS_1:th.14">PCOMPS_1:14</a></i>;<br/><a NAME="E16:7_1"/>
(  <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">clf</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( for <font color="Olive" title="b1">Gd</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Gd</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">Gd</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">clf</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Olive" title="b1">Gd</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> holds <br/> <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Olive" title="b1">Gd</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:7_1_3"/>
 <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">clf</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="uniform1.html#E14:7_1"><i><font color="Green" title="E16">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Gd</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Gd</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">Gd</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">clf</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Olive" title="b1">Gd</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> holds <br/> <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Olive" title="b1">Gd</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<b>let </b><font color="Maroon" title="c10">H</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">H</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c10">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">clf</a> <font color="Maroon" title="c7">F</font> &amp; <font color="Maroon" title="c10">H</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> implies  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c10">H</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:7_1_3"/><i><font color="Green" title="E18">A13</font></i>: 
<font color="Maroon" title="c10">H</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>and </b><br/><a NAME="E3:7_1_3"/><i><font color="Green" title="E19">A14</font></i>: 
<font color="Maroon" title="c10">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">clf</a> <font color="Maroon" title="c7">F</font>
 <b>and </b><br/><a NAME="E4:7_1_3"/><i><font color="Green" title="E20">A15</font></i>: 
<font color="Maroon" title="c10">H</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c10">H</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<b>reconsider </b><font color="Maroon" title="c11">H</font> = <font color="Maroon" title="c10">H</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span> <b>by </b><i><a class="txt" href="uniform1.html#E3:7_1_3"><i><font color="Green" title="E19">A14</font></i></a>, <a class="ref" href="tops_2.html#T3" title="TOPS_2:th.3">TOPS_2:3</a></i>;<br/>
<a NAME="E6:7_1_3"/>
<font color="Maroon" title="c11">H</font> is <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">B</font>, <font color="Maroon" title="c13">C</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal1.html#D9" title="ORDINAL1:def.9">ORDINAL1:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c12">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">H</font> or  not <font color="Maroon" title="c13">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">H</font> or  not <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">R7</a>(<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c13">C</font>) )</span><br/>

<b>assume </b><a NAME="E1:7_1_3_1"/><i><font color="Green" title="E21">A16</font></i>: 
( <font color="Maroon" title="c12">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">H</font> &amp; <font color="Maroon" title="c13">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">H</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">R7</a>(<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c13">C</font>)</span><br/>

<b>then reconsider </b><font color="Maroon" title="c14">B</font> = <font color="Maroon" title="c12">B</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#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c15">C</font> = <font color="Maroon" title="c13">C</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#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span> <b>by </b><i><a class="txt" href="uniform1.html#E1:7_1_3_1"><i><font color="Green" title="E21">A16</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">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#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span><b> such that </b><br/><a NAME="E5:7_1_3_1"/><i><font color="Green" title="E22">A17</font></i>: 
( <font color="Maroon" title="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c16">V</font> &amp; <font color="Maroon" title="c16">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F</font> )
 <b>by </b><i><a class="txt" href="uniform1.html#E3:7_1_3"><i><font color="Green" title="E19">A14</font></i></a>, <a class="txt" href="uniform1.html#E1:7_1_3_1"><i><font color="Green" title="E21">A16</font></i></a>, <a class="ref" href="pcomps_1.html#D3" title="PCOMPS_1:def.3">PCOMPS_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">W</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#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span><b> such that </b><br/><a NAME="E7:7_1_3_1"/><i><font color="Green" title="E23">A18</font></i>: 
( <font color="Maroon" title="c15">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c17">W</font> &amp; <font color="Maroon" title="c17">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F</font> )
 <b>by </b><i><a class="txt" href="uniform1.html#E3:7_1_3"><i><font color="Green" title="E19">A14</font></i></a>, <a class="txt" href="uniform1.html#E1:7_1_3_1"><i><font color="Green" title="E21">A16</font></i></a>, <a class="ref" href="pcomps_1.html#D3" title="PCOMPS_1:def.3">PCOMPS_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">p</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E9:7_1_3_1"/><i><font color="Green" title="E24">A19</font></i>: 
<font color="Maroon" title="c16">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">N</font> :  ex <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c18">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="uniform1.html#E8:7_1"><i><font color="Green" title="E12">A7</font></i></a>, <a class="txt" href="uniform1.html#E5:7_1_3_1"><i><font color="Green" title="E22">A17</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c19">q</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E11:7_1_3_1"/><i><font color="Green" title="E25">A20</font></i>: 
<font color="Maroon" title="c17">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">N</font> :  ex <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c19">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">n</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="uniform1.html#E8:7_1"><i><font color="Green" title="E12">A7</font></i></a>, <a class="txt" href="uniform1.html#E7:7_1_3_1"><i><font color="Green" title="E23">A18</font></i></a></i>;<br/>
<a NAME="E12:7_1_3_1"/>
( <font color="Maroon" title="c16">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">W</font> or <font color="Maroon" title="c17">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">V</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_3_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_3_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_3_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_1_3_1_1_1_1"/>
( <font color="Maroon" title="c19">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">p</font> or <font color="Maroon" title="c18">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">q</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_3_1_1_1_1_1"><b>case </b></a><a NAME="E1:7_1_3_1_1_1_1_1"/><i><font color="Green" title="E26">A21</font></i>: 
<font color="Maroon" title="c19">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c16">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">W</font></span><br/><div class="add"><b>thus </b><a NAME="E2:7_1_3_1_1_1_1_1"/>
<font color="Maroon" title="c16">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">W</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_3_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">V</font> or <font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">W</font> )</span><br/>

<b>assume </b><a NAME="E1:7_1_3_1_1_1_1_1_1"/>
<font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">V</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">W</font></span><br/>

<b>then consider </b><font color="Maroon" title="c21">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">N</font><b> such that </b><br/><a NAME="E3:7_1_3_1_1_1_1_1_1"/><i><font color="Green" title="E27">A22</font></i>: 
( <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">x</font> &amp;  ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c18">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> ) )
 <b>by </b><i><a class="txt" href="uniform1.html#E9:7_1_3_1"><i><font color="Green" title="E24">A19</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c22">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:7_1_3_1_1_1_1_1_1"/><i><font color="Green" title="E28">A23</font></i>: 
( <font color="Maroon" title="c18">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">n</font> &amp; <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">n</font> )
 <b>by </b><i><a class="txt" href="uniform1.html#E3:7_1_3_1_1_1_1_1_1"><i><font color="Green" title="E27">A22</font></i></a></i>;<br/>
<a NAME="E6:7_1_3_1_1_1_1_1_1"/>
( <font color="Maroon" title="c19">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">n</font> &amp; <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">n</font> )
 
<b>by </b><i><a class="txt" href="uniform1.html#E1:7_1_3_1_1_1_1_1"><i><font color="Green" title="E26">A21</font></i></a>, <a class="txt" href="uniform1.html#E5:7_1_3_1_1_1_1_1_1"><i><font color="Green" title="E28">A23</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E7:7_1_3_1_1_1_1_1_1"/>
<font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">W</font>
 <b>by </b><i><a class="txt" href="uniform1.html#E11:7_1_3_1"><i><font color="Green" title="E25">A20</font></i></a>, <a class="txt" href="uniform1.html#E3:7_1_3_1_1_1_1_1_1"><i><font color="Green" title="E27">A22</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_3_1_1_1_1_2"><b>case </b></a><a NAME="E1:7_1_3_1_1_1_1_2"/><i><font color="Green" title="E26">A24</font></i>: 
<font color="Maroon" title="c18">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c17">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">V</font></span><br/><div class="add"><b>thus </b><a NAME="E2:7_1_3_1_1_1_1_2"/>
<font color="Maroon" title="c17">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">V</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_3_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">W</font> or <font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">V</font> )</span><br/>

<b>assume </b><a NAME="E1:7_1_3_1_1_1_1_2_1"/>
<font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">V</font></span><br/>

<b>then consider </b><font color="Maroon" title="c21">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">N</font><b> such that </b><br/><a NAME="E3:7_1_3_1_1_1_1_2_1"/><i><font color="Green" title="E27">A25</font></i>: 
( <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">x</font> &amp;  ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c19">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> ) )
 <b>by </b><i><a class="txt" href="uniform1.html#E11:7_1_3_1"><i><font color="Green" title="E25">A20</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c22">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:7_1_3_1_1_1_1_2_1"/><i><font color="Green" title="E28">A26</font></i>: 
( <font color="Maroon" title="c19">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">n</font> &amp; <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">n</font> )
 <b>by </b><i><a class="txt" href="uniform1.html#E3:7_1_3_1_1_1_1_2_1"><i><font color="Green" title="E27">A25</font></i></a></i>;<br/>
<a NAME="E6:7_1_3_1_1_1_1_2_1"/>
( <font color="Maroon" title="c18">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">n</font> &amp; <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">n</font> )
 
<b>by </b><i><a class="txt" href="uniform1.html#E1:7_1_3_1_1_1_1_2"><i><font color="Green" title="E26">A24</font></i></a>, <a class="txt" href="uniform1.html#E5:7_1_3_1_1_1_1_2_1"><i><font color="Green" title="E28">A26</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E7:7_1_3_1_1_1_1_2_1"/>
<font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">V</font>
 <b>by </b><i><a class="txt" href="uniform1.html#E9:7_1_3_1"><i><font color="Green" title="E24">A19</font></i></a>, <a class="txt" href="uniform1.html#E3:7_1_3_1_1_1_1_2_1"><i><font color="Green" title="E27">A25</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:7_1_3_1_1"/>
( <font color="Maroon" title="c16">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">W</font> or <font color="Maroon" title="c17">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">V</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E13:7_1_3_1"/><b>then </b>
( <font color="Maroon" title="c14">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">C</font> or <font color="Maroon" title="c15">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">B</font> )
 
<b>by </b><i><a class="txt" href="uniform1.html#E5:7_1_3_1"><i><font color="Green" title="E22">A17</font></i></a>, <a class="txt" href="uniform1.html#E7:7_1_3_1"><i><font color="Green" title="E23">A18</font></i></a>, <a class="ref" href="pre_topc.html#T49" title="PRE_TOPC:th.49">PRE_TOPC:49</a></i>;<br/>
<b>hence </b><a NAME="E14:7_1_3_1"/>
<a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">R7</a>(<font color="Maroon" title="c12">B</font>,<font color="Maroon" title="c13">C</font>)
 <b>by </b><i><a class="ref" href="xboole_0.html#D9" title="XBOOLE_0:def.9">XBOOLE_0:def 9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c12">m</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:7_1_3"/><i><font color="Green" title="E21">A27</font></i>: 
<font color="Maroon" title="c12">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">H</font>
 <b>and </b><br/><a NAME="E9:7_1_3"/><i><font color="Green" title="E22">A28</font></i>: 
for <font color="Olive" title="b1">C</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">H</font> holds <br/><font color="Maroon" title="c12">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">C</font>
 <b>by </b><i><a class="txt" href="uniform1.html#E2:7_1_3"><i><font color="Green" title="E18">A13</font></i></a>, <a class="txt" href="uniform1.html#E4:7_1_3"><i><font color="Green" title="E20">A15</font></i></a>, <a class="ref" href="finset_1.html#T30" title="FINSET_1:th.30">FINSET_1:30</a></i>;<br/>
<a NAME="E10:7_1_3"/><i><font color="Green" title="E23">A29</font></i>: 
<font color="Maroon" title="c12">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c11">H</font>
 
<b>by </b><i><a class="txt" href="uniform1.html#E2:7_1_3"><i><font color="Green" title="E18">A13</font></i></a>, <a class="txt" href="uniform1.html#E9:7_1_3"><i><font color="Green" title="E22">A28</font></i></a>, <a class="ref" href="setfam_1.html#T6" title="SETFAM_1:th.6">SETFAM_1:6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">m</font> = <font color="Maroon" title="c12">m</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#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span> <b>by </b><i><a class="txt" href="uniform1.html#E8:7_1_3"><i><font color="Green" title="E21">A27</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">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="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span><b> such that </b><br/><a NAME="E13:7_1_3"/><i><font color="Green" title="E24">A30</font></i>: 
( <font color="Maroon" title="c13">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c14">A</font> &amp; <font color="Maroon" title="c14">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F</font> )
 <b>by </b><i><a class="txt" href="uniform1.html#E3:7_1_3"><i><font color="Green" title="E19">A14</font></i></a>, <a class="txt" href="uniform1.html#E8:7_1_3"><i><font color="Green" title="E21">A27</font></i></a>, <a class="ref" href="pcomps_1.html#D3" title="PCOMPS_1:def.3">PCOMPS_1:def 3</a></i>;<br/>
<a NAME="E14:7_1_3"/>
<font color="Maroon" title="c14">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="uniform1.html#E8:7_1"><i><font color="Green" title="E12">A7</font></i></a>, <a class="txt" href="uniform1.html#E9:7_1"><i><font color="Green" title="E13">A8</font></i></a>, <a class="txt" href="uniform1.html#E13:7_1_3"><i><font color="Green" title="E24">A30</font></i></a></i>;<br/>
<a NAME="E15:7_1_3"/><b>then </b>
<font color="Maroon" title="c13">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="uniform1.html#E13:7_1_3"><i><font color="Green" title="E24">A30</font></i></a>, <a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/>
<b>hence </b><a NAME="E16:7_1_3"/>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c10">H</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="uniform1.html#E10:7_1_3"><i><font color="Green" title="E23">A29</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E17:7_1"/><b>then </b>
 <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">clf</a> <font color="Maroon" title="c7">F</font> is <a href="compts_1.html#V1" title="COMPTS_1:attr.1">centered</a>
 <b>by </b><i><a class="ref" href="compts_1.html#D2" title="COMPTS_1:def.2">COMPTS_1:def 2</a></i>;<br/><a NAME="E18:7_1"/><b>then </b>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">clf</a> <font color="Maroon" title="c7">F</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="uniform1.html#E1:7"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="uniform1.html#E15:7_1"><i><font color="Green" title="E17">A12</font></i></a>, <a class="ref" href="compts_1.html#T13" title="COMPTS_1:th.13">COMPTS_1:13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">c</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span><b> such that </b><br/><a NAME="E20:7_1"/><i><font color="Green" title="E18">A31</font></i>: 
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">clf</a> <font color="Maroon" title="c7">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">c</font> = <font color="Maroon" title="c10">c</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">N</font> <b>by </b><i><a class="txt" href="uniform1.html#E6:7_1"><i><font color="Green" title="E11">A6</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c12">Ge</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#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span><b> such that </b><br/><a NAME="E23:7_1"/><i><font color="Green" title="E19">A32</font></i>: 
( <font color="Maroon" title="c11">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">Ge</font> &amp; <font color="Maroon" title="c12">Ge</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> )
 <b>by </b><i><a class="txt" href="uniform1.html#E1:7"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="pcomps_1.html#T5" title="PCOMPS_1:th.5">PCOMPS_1:5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">Ge</font> = <font color="Maroon" title="c12">Ge</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#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span> ;<br/><a NAME="E25:7_1"/>
<font color="Maroon" title="c13">Ge</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="uniform1.html#E1:7"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="uniform1.html#E23:7_1"><i><font color="Green" title="E19">A32</font></i></a>, <a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c14">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> <b> such that </b><br/><a NAME="E27:7_1"/><i><font color="Green" title="E20">A33</font></i>: 
( <font color="Maroon" title="c14">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>  &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c14">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">Ge</font> )
 <b>by </b><i><a class="txt" href="uniform1.html#E23:7_1"><i><font color="Green" title="E19">A32</font></i></a>, <a class="ref" href="topmetr.html#T22" title="TOPMETR:th.22">TOPMETR:22</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">r</font> = <font color="Maroon" title="c14">r</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><a NAME="E29:7_1"/><i><font color="Green" title="E21">A34</font></i>: 
<font color="Maroon" title="c15">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="uniform1.html#E27:7_1"><i><font color="Green" title="E20">A33</font></i></a>, <a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c16">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E31:7_1"/><i><font color="Green" title="E22">A35</font></i>: 
( <font color="Maroon" title="c16">n</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>  &amp; 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c15">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 <b>by </b><i><a class="ref" href="uniform1.html#T2" target="_self" title="UNIFORM1:th.2">Th2</a></i>;<br/><a NAME="E32:7_1"/>
<font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E33:7_1"/><b>then </b>
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="uniform1.html#E31:7_1"><i><font color="Green" title="E22">A35</font></i></a>, <a class="ref" href="real_2.html#T152" title="REAL_2:th.152">REAL_2:152</a></i>;<br/><a NAME="E34:7_1"/><b>then </b><i><font color="Green" title="E23">A36</font></i>: 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c15">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <b>by </b><i><a class="txt" href="uniform1.html#E31:7_1"><i><font color="Green" title="E22">A35</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">n2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n2</font> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n2</font> );<br/><b>reconsider </b><font color="Maroon" title="c17">B2</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>] </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="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span> <b>from </b><i><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></i><a NAME="E36:7_1"/><i><font color="Green" title="E24">A37</font></i>: 
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#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">N</font>
 <b>by </b><i><a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></i>;<br/><a NAME="E37:7_1"/><b>then </b>
<font color="Maroon" title="c17">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">F</font>
 <b>by </b><i><a class="txt" href="uniform1.html#E8:7_1"><i><font color="Green" title="E12">A7</font></i></a></i>;<br/><a NAME="E38:7_1"/><b>then </b>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c17">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">clf</a> <font color="Maroon" title="c7">F</font>
 <b>by </b><i><a class="ref" href="pcomps_1.html#D3" title="PCOMPS_1:def.3">PCOMPS_1:def 3</a></i>;<br/><a NAME="E39:7_1"/><b>then </b><i><font color="Green" title="E25">A38</font></i>: 
<font color="Maroon" title="c11">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c17">B2</font>
 <b>by </b><i><a class="txt" href="uniform1.html#E20:7_1"><i><font color="Green" title="E18">A31</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">Q1</font> =  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</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="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></i>;<br/><a NAME="E41:7_1"/><i><font color="Green" title="E26">A39</font></i>: 
<font color="Maroon" title="c11">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Q1</font>
 <b>by </b><i><a class="txt" href="uniform1.html#E29:7_1"><i><font color="Green" title="E21">A34</font></i></a>, <a class="ref" href="goboard6.html#T4" title="GOBOARD6:th.4">GOBOARD6:4</a></i>;<br/><a NAME="E42:7_1"/>
<font color="Maroon" title="c18">Q1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="topmetr.html#T21" title="TOPMETR:th.21">TOPMETR:21</a></i>;<br/><a NAME="E43:7_1"/><b>then </b>
<font color="Maroon" title="c17">B2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c18">Q1</font>
 <b>by </b><i><a class="txt" href="uniform1.html#E39:7_1"><i><font color="Green" title="E25">A38</font></i></a>, <a class="txt" href="uniform1.html#E41:7_1"><i><font color="Green" title="E26">A39</font></i></a>, <a class="ref" href="tops_1.html#T39" title="TOPS_1:th.39">TOPS_1:39</a></i>;<br/><b>then consider </b><font color="Maroon" title="c19">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E45:7_1"/><i><font color="Green" title="E27">A40</font></i>: 
( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">B2</font> &amp; <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Q1</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>consider </b><font color="Maroon" title="c20">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">N</font><b> such that </b><br/><a NAME="E47:7_1"/><i><font color="Green" title="E28">A41</font></i>: 
( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">y</font> &amp;  ex <font color="Olive" title="b1">n2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n2</font> &amp; <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n2</font> ) )
 <b>by </b><i><a class="txt" href="uniform1.html#E36:7_1"><i><font color="Green" title="E24">A37</font></i></a>, <a class="txt" href="uniform1.html#E45:7_1"><i><font color="Green" title="E27">A40</font></i></a></i>;<br/><a NAME="E48:7_1"/><i><font color="Green" title="E29">A42</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c20">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c15">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <b>by </b><i><a class="txt" href="uniform1.html#E45:7_1"><i><font color="Green" title="E27">A40</font></i></a>, <a class="txt" href="uniform1.html#E47:7_1"><i><font color="Green" title="E28">A41</font></i></a>, <a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/><a NAME="E49:7_1"/>
<font color="Maroon" title="c15">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c15">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <b>by </b><i><a class="txt" href="uniform1.html#E27:7_1"><i><font color="Green" title="E20">A33</font></i></a>, <a class="ref" href="real_2.html#T200" title="REAL_2:th.200">REAL_2:200</a></i>;<br/><a NAME="E50:7_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c20">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c15">r</font>
 <b>by </b><i><a class="txt" href="uniform1.html#E48:7_1"><i><font color="Green" title="E29">A42</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E51:7_1"/><b>then </b><i><font color="Green" title="E30">A43</font></i>: 
<font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c15">r</font>
 <b>by </b><i><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/><b>consider </b><font color="Maroon" title="c21">n2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E53:7_1"/><i><font color="Green" title="E31">A44</font></i>: 
( <font color="Maroon" title="c16">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">n2</font> &amp; <font color="Maroon" title="c20">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">n2</font> )
 <b>by </b><i><a class="txt" href="uniform1.html#E47:7_1"><i><font color="Green" title="E28">A41</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c22">w2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font><b> such that </b><br/><a NAME="E55:7_1"/><i><font color="Green" title="E32">A45</font></i>: 
(  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c20">y</font>,<font color="Maroon" title="c22">w2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; ( for <font color="Olive" title="b1">Ga</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#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span>  holds <br/>(  not <font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Ga</font> or  not <font color="Maroon" title="c22">w2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Ga</font> or  not <font color="Olive" title="b1">Ga</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> ) ) )
 <b>by </b><i><a class="txt" href="uniform1.html#E5:7_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="uniform1.html#E53:7_1"><i><font color="Green" title="E31">A44</font></i></a></i>;<br/><a NAME="E56:7_1"/>
<font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="uniform1.html#E53:7_1"><i><font color="Green" title="E31">A44</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E57:7_1"/><b>then </b>
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="real_2.html#T201" title="REAL_2:th.201">REAL_2:201</a></i>;<br/><a NAME="E58:7_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c20">y</font>,<font color="Maroon" title="c22">w2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="uniform1.html#E55:7_1"><i><font color="Green" title="E32">A45</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E59:7_1"/><b>then </b><i><font color="Green" title="E33">A46</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c20">y</font>,<font color="Maroon" title="c22">w2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c15">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <b>by </b><i><a class="txt" href="uniform1.html#E34:7_1"><i><font color="Green" title="E23">A36</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E60:7_1"/><i><font color="Green" title="E34">A47</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c22">w2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c20">y</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c20">y</font>,<font color="Maroon" title="c22">w2</font></span>)</span>
 <b>by </b><i><a class="ref" href="metric_1.html#T4" title="METRIC_1:th.4">METRIC_1:4</a></i>;<br/><a NAME="E61:7_1"/><i><font color="Green" title="E35">A48</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c20">y</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c20">y</font>,<font color="Maroon" title="c22">w2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c20">y</font>,<font color="Maroon" title="c22">w2</font></span>)</span>
 <b>by </b><i><a class="txt" href="uniform1.html#E48:7_1"><i><font color="Green" title="E29">A42</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E62:7_1"/><i><font color="Green" title="E36">A49</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c20">y</font>,<font color="Maroon" title="c22">w2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="uniform1.html#E59:7_1"><i><font color="Green" title="E33">A46</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E63:7_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c22">w2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c20">y</font>,<font color="Maroon" title="c22">w2</font></span>)</span>
 <b>by </b><i><a class="txt" href="uniform1.html#E60:7_1"><i><font color="Green" title="E34">A47</font></i></a>, <a class="txt" href="uniform1.html#E61:7_1"><i><font color="Green" title="E35">A48</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E64:7_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c22">w2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">r</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="uniform1.html#E62:7_1"><i><font color="Green" title="E36">A49</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E65:7_1"/><b>then </b>
<font color="Maroon" title="c22">w2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c15">r</font>
 <b>by </b><i><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/><b>hence </b><a NAME="E66:7_1"/>
contradiction
 <b>by </b><i><a class="txt" href="uniform1.html#E23:7_1"><i><font color="Green" title="E19">A32</font></i></a>, <a class="txt" href="uniform1.html#E27:7_1"><i><font color="Green" title="E20">A33</font></i></a>, <a class="txt" href="uniform1.html#E51:7_1"><i><font color="Green" title="E30">A43</font></i></a>, <a class="txt" href="uniform1.html#E55:7_1"><i><font color="Green" title="E32">A45</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:7"/>
 ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">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>  &amp; ( for <font color="Olive" title="b2">w1</font>, <font color="Olive" title="b3">w2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font>  st  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Olive" title="b2">w1</font>,<font color="Olive" title="b3">w2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">r</font> holds <br/> ex <font color="Olive" title="b4">Ga</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#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span> st <br/>( <font color="Olive" title="b2">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">Ga</font> &amp; <font color="Olive" title="b3">w2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">Ga</font> &amp; <font color="Olive" title="b4">Ga</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> ) ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
