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

<b>let </b><font color="Maroon" title="c1">GX</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">W</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">GX</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:21"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c2">W</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 <b>and </b><br/><a NAME="E2:21"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c2">W</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/>

<b>consider </b><font color="Maroon" title="c3">p</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E4:21"/><i><font color="Green" title="E13">A3</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font>
 <b>by </b><i><a class="txt" href="tops_2.html#E2:21"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="finseq_1.html#T73" title="FINSEQ_1:th.73">FINSEQ_1:73</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E6:21"/><i><font color="Green" title="E14">A4</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>for <font color="Olive" title="b1">Z</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">GX</font>  st <font color="Olive" title="b1">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> $1</span>)</span> &amp; $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> holds <br/> <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">Z</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>;<br/>
<a NAME="E7:21"/><i><font color="Green" title="E15">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">Z</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">GX</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:21_1"/><i><font color="Green" title="E16">A6</font></i>: 
<font color="Maroon" title="c5">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <b>and </b><br/><a NAME="E2:21_1"/>
 <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="Maroon" title="c4">n</font>
 ;<br/>

<a NAME="E3:21_1"/><i><font color="Green" title="E17">A7</font></i>: 
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c5">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="tops_2.html#E1:21_1"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="relat_1.html#T149" title="RELAT_1:th.149">RELAT_1:149</a>, <a class="ref" href="setfam_1.html#T2" title="SETFAM_1:th.2">SETFAM_1:2</a></i>;<br/>
<a NAME="E4:21_1"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">GX</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T5" title="PRE_TOPC:th.5">PRE_TOPC:5</a></i>;<br/>
<b>hence </b><a NAME="E5:21_1"/>
(  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> or  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c5">Z</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <b>by </b><i><a class="txt" href="tops_2.html#E3:21_1"><i><font color="Green" title="E17">A7</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:21"/><i><font color="Green" title="E16">A8</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:21_2"/><i><font color="Green" title="E17">A9</font></i>: 
for <font color="Olive" title="b1">Z</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">GX</font>  st <font color="Olive" title="b1">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">k</font></span>)</span> &amp; <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> holds <br/> <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">Z</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c6">Z</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">GX</font>;<br/>

<b>assume </b><a NAME="E2:21_2"/><i><font color="Green" title="E18">A10</font></i>: 
<font color="Maroon" title="c6">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 ;<br/>

<b>assume </b><a NAME="E3:21_2"/><i><font color="Green" title="E19">A11</font></i>: 
( <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> )
 ;<br/>

<div><i><font color="Green" title="E20">A12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_2_1"/><i><font color="Green" title="E21">A13</font></i>: 
<font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E2:21_2_1"/><b>then </b><i><font color="Green" title="E22">A14</font></i>: 
(  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">p</font> )
 <b>by </b><i><a class="txt" href="tops_2.html#E6:21"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="tops_2.html#E3:21_2"><i><font color="Green" title="E19">A11</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a></i>;<br/><a NAME="E3:21_2_1"/><b>then </b>
 <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c3">p</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="funct_1.html#T117" title="FUNCT_1:th.117">FUNCT_1:117</a></i>;<br/><a NAME="E4:21_2_1"/><b>then </b>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c6">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <b>by </b><i><a class="txt" href="tops_2.html#E2:21_2"><i><font color="Green" title="E18">A10</font></i></a>, <a class="txt" href="tops_2.html#E1:21_2_1"><i><font color="Green" title="E21">A13</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="setfam_1.html#T11" title="SETFAM_1:th.11">SETFAM_1:11</a></i>;<br/><a NAME="E5:21_2_1"/><b>then </b>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c6">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W</font>
 <b>by </b><i><a class="txt" href="tops_2.html#E4:21"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="tops_2.html#E2:21_2_1"><i><font color="Green" title="E22">A14</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E6:21_2_1"/>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c6">Z</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="tops_2.html#E1:21"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="tops_2.html#D1" target="_self" title="TOPS_2:def.1">Def1</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_2_2"/><i><font color="Green" title="E21">A15</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k</font>
 ;<br/><a NAME="E2:21_2_2"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E3:21_2_2"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 <b>by </b><i><a class="txt" href="tops_2.html#E1:21_2_2"><i><font color="Green" title="E21">A15</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:21_2_2"/><b>then </b><i><font color="Green" title="E22">A16</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">k</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="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:21_2_2"/>
<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="tops_2.html#E3:21_2"><i><font color="Green" title="E19">A11</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E6:21_2_2"/><b>then </b>
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E7:21_2_2"/><b>then </b><i><font color="Green" title="E23">A17</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="txt" href="tops_2.html#E6:21"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a></i>;<br/><i><font color="Green" title="E24">A18</font></i>: <font color="Maroon" title="c3">p</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> = 
<font color="Maroon" title="c3">p</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T11" title="FINSEQ_1:th.11">FINSEQ_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T153" title="RELAT_1:th.153">RELAT_1:153</a></i>
;<br/><a NAME="E9:21_2_2"/>
<font color="Maroon" title="c3">p</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">W</font>
 <b>by </b><i><a class="txt" href="tops_2.html#E4:21"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="relat_1.html#T144" title="RELAT_1:th.144">RELAT_1:144</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c7">G1</font> = <font color="Maroon" title="c3">p</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">k</font></span>)</span> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">GX</font> <b>by </b><i><a class="ref" href="tops_2.html#T3" target="_self" title="TOPS_2:th.3">Th3</a></i>;<br/><a NAME="E11:21_2_2"/><i><font color="Green" title="E25">A19</font></i>: 
<font color="Maroon" title="c7">G1</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="tops_2.html#E4:21_2_2"><i><font color="Green" title="E22">A16</font></i></a>, <a class="txt" href="tops_2.html#E7:21_2_2"><i><font color="Green" title="E23">A17</font></i></a>, <a class="ref" href="relat_1.html#T152" title="RELAT_1:th.152">RELAT_1:152</a></i>;<br/><a NAME="E12:21_2_2"/>
 <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">W</font>
 <b>by </b><i><a class="txt" href="tops_2.html#E4:21"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="relat_1.html#T144" title="RELAT_1:th.144">RELAT_1:144</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">G2</font> =  <a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c3">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">GX</font> <b>by </b><i><a class="ref" href="tops_2.html#T3" target="_self" title="TOPS_2:th.3">Th3</a></i>;<br/><a NAME="E14:21_2_2"/>
(  <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="Maroon" title="c5">k</font> &amp; <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></i>;<br/><a NAME="E15:21_2_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> )
 <b>by </b><i><a class="txt" href="tops_2.html#E3:21_2"><i><font color="Green" title="E19">A11</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E16:21_2_2"/><b>then </b><i><font color="Green" title="E26">A20</font></i>: 
<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="txt" href="tops_2.html#E6:21"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E17:21_2_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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>  &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">p</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><a NAME="E18:21_2_2"/><b>then </b>
<font color="Maroon" title="c8">G2</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="ref" href="relat_1.html#T152" title="RELAT_1:th.152">RELAT_1:152</a></i>;<br/><a NAME="E19:21_2_2"/><b>then </b><i><font color="Green" title="E27">A21</font></i>: 
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c6">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c7">G1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c8">G2</font></span>)</span>
 <b>by </b><i><a class="txt" href="tops_2.html#E2:21_2"><i><font color="Green" title="E18">A10</font></i></a>, <a class="txt" href="tops_2.html#E8:21_2_2"><i><font color="Green" title="E24">A18</font></i></a>, <a class="txt" href="tops_2.html#E11:21_2_2"><i><font color="Green" title="E25">A19</font></i></a>, <a class="ref" href="setfam_1.html#T10" title="SETFAM_1:th.10">SETFAM_1:10</a></i>;<br/><a NAME="E20:21_2_2"/>
<font color="Maroon" title="c8">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="tops_2.html#E16:21_2_2"><i><font color="Green" title="E26">A20</font></i></a>, <a class="ref" href="funct_1.html#T117" title="FUNCT_1:th.117">FUNCT_1:117</a></i>;<br/><a NAME="E21:21_2_2"/><b>then </b><i><font color="Green" title="E28">A22</font></i>: 
(  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c8">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W</font> )
 <b>by </b><i><a class="txt" href="tops_2.html#E4:21"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="tops_2.html#E16:21_2_2"><i><font color="Green" title="E26">A20</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="ref" href="setfam_1.html#T11" title="SETFAM_1:th.11">SETFAM_1:11</a></i>;<br/><a NAME="E22:21_2_2"/>
<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="tops_2.html#E3:21_2"><i><font color="Green" title="E19">A11</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E23:21_2_2"/><b>then </b>
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E24:21_2_2"/><b>then </b>
(  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c7">G1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp;  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c8">G2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <b>by </b><i><a class="txt" href="tops_2.html#E1:21"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="tops_2.html#E1:21_2"><i><font color="Green" title="E17">A9</font></i></a>, <a class="txt" href="tops_2.html#E3:21_2"><i><font color="Green" title="E19">A11</font></i></a>, <a class="txt" href="tops_2.html#E21:21_2_2"><i><font color="Green" title="E28">A22</font></i></a>, <a class="ref" href="tops_2.html#D1" target="_self" title="TOPS_2:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E25:21_2_2"/>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c6">Z</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="tops_2.html#E19:21_2_2"><i><font color="Green" title="E27">A21</font></i></a>, <a class="ref" href="tops_1.html#T38" title="TOPS_1:th.38">TOPS_1:38</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:21_2"/>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c6">Z</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="tops_2.html#E4:21_2"><i><font color="Green" title="E20">A12</font></i></a>, <a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:21"/><i><font color="Green" title="E17">A23</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="tops_2.html#E7:21"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="tops_2.html#E8:21"><i><font color="Green" title="E16">A8</font></i></a>);<br/></i>
<div><i><font color="Green" title="E18">A24</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_3"/><i><font color="Green" title="E19">A25</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 ;<br/><a NAME="E2:21_3"/>
<font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="tops_2.html#E4:21"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="tops_2.html#E6:21"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="relat_1.html#T146" title="RELAT_1:th.146">RELAT_1:146</a></i>;<br/><b>hence </b><a NAME="E3:21_3"/>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c2">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="tops_2.html#E9:21"><i><font color="Green" title="E17">A23</font></i></a>, <a class="txt" href="tops_2.html#E1:21_3"><i><font color="Green" title="E19">A25</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E19">A26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_4"/>
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E2:21_4"/><b>then </b>
<font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="tops_2.html#E4:21"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="tops_2.html#E6:21"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="relat_1.html#T146" title="RELAT_1:th.146">RELAT_1:146</a></i>;<br/><a NAME="E3:21_4"/><b>then </b><i><font color="Green" title="E20">A27</font></i>: 
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#T149" title="RELAT_1:th.149">RELAT_1:149</a>, <a class="ref" href="setfam_1.html#T2" title="SETFAM_1:th.2">SETFAM_1:2</a></i>;<br/><a NAME="E4:21_4"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">GX</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T5" title="PRE_TOPC:th.5">PRE_TOPC:5</a></i>;<br/><b>hence </b><a NAME="E5:21_4"/>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c2">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="tops_2.html#E3:21_4"><i><font color="Green" title="E20">A27</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_5"/>
<font color="Maroon" title="c4">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E2:21_5"/><b>then </b>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">n</font> &amp; 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></i>;<br/><b>hence </b><a NAME="E3:21_5"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:21"/>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c2">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="tops_2.html#E10:21"><i><font color="Green" title="E18">A24</font></i></a>, <a class="txt" href="tops_2.html#E11:21"><i><font color="Green" title="E19">A26</font></i></a></i>;<br/>


</div>
