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

<span class="kw">let </span><font color="Maroon" title="c1">GX</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">W</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">W</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  &amp; <font color="Olive" title="b1">W</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  holds <br/> <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Olive" title="b1">W</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">W</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>  &amp; <font color="Maroon" title="c2">W</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  implies  <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>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:28"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
<font color="Maroon" title="c2">W</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> 
 <span class="kw">and </span><br/><a NAME="E2:28"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
<font color="Maroon" title="c2">W</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c3">p</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E4:28"/><span class="lab"><font color="Green" title="E13">A3</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="finseq_1.html#T73" title="FINSEQ_1:th.73">FINSEQ_1:73</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:28"/><span class="lab"><font color="Green" title="E14">A4</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>for <font color="Olive" title="b1">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="E7:28"/><span class="lab"><font color="Green" title="E15">A5</font></span>: 
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="28_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:28_1"/><span class="lab"><font color="Green" title="E16">A6</font></span>: 
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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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> &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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> implies  <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>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:28_1"/><span class="lab"><font color="Green" title="E17">A7</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <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> or  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="c6">Z</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:28_1"/><span class="lab"><font color="Green" title="E18">A8</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:28_1"/><span class="lab"><font color="Green" title="E19">A9</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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> </span><br/>

<div><span class="lab"><font color="Green" title="E20">A10</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c7">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="tops_2.html#T3" target="_self" title="TOPS_2:th.3">Th3</a>, <a class="ref" href="relat_1.html#T144" title="RELAT_1:th.144">RELAT_1:144</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="tops_2.html#T3" target="_self" title="TOPS_2:th.3">Th3</a>, <a class="ref" href="relat_1.html#T144" title="RELAT_1:th.144">RELAT_1:144</a></span>;<br/><span class="kw">assume </span><a NAME="E3:28_1_1"/><span class="lab"><font color="Green" title="E21">A12</font></span>: 
 <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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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> </span><br/><a NAME="E4:28_1_1"/>
<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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_1"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E5:28_1_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E6:28_1_1"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a></span>;<br/><a NAME="E7:28_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A13</font></span>: 
<font color="Maroon" title="c8">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_1_1"><span class="lab"><font color="Green" title="E21">A12</font></span></a>, <a class="ref" href="relat_1.html#T152" title="RELAT_1:th.152">RELAT_1:152</a></span>;<br/><a NAME="E8:28_1_1"/>
<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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_1"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E9:28_1_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E10:28_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A14</font></span>: 
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c8">G1</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="txt" href="#E4:28_1"><span class="lab"><font color="Green" title="E19">A9</font></span></a></span>;<br/><a NAME="E11:28_1_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="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 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></span>;<br/><a NAME="E12:28_1_1"/><span class="kw">then </span>
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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E13:28_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A15</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="txt" href="#E3:28_1"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E14:28_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T117" title="FUNCT_1:th.117">FUNCT_1:117</a></span>;<br/><a NAME="E15:28_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A16</font></span>: 
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c7">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#T11" title="SETFAM_1:th.11">SETFAM_1:11</a></span>;<br/><a NAME="E16:28_1_1"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:28_1_1"><span class="lab"><font color="Green" title="E24">A15</font></span></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/><a NAME="E17:28_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A17</font></span>: 
<font color="Maroon" title="c7">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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T152" title="RELAT_1:th.152">RELAT_1:152</a></span>;<br/><a NAME="E18:28_1_1"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="txt" href="#E13:28_1_1"><span class="lab"><font color="Green" title="E24">A15</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E19:28_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A18</font></span>: 
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c7">G2</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E15:28_1_1"><span class="lab"><font color="Green" title="E25">A16</font></span></a>, <a class="ref" href="tops_2.html#D1" target="_self" title="TOPS_2:def.1">Def1</a></span>;<br/><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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T11" title="FINSEQ_1:th.11">FINSEQ_1:11</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T153" title="RELAT_1:th.153">RELAT_1:153</a></span>
;<br/><a NAME="E21:28_1_1"/><span class="kw">then </span>
 <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="c8">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="c7">G2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_1"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="txt" href="#E7:28_1_1"><span class="lab"><font color="Green" title="E22">A13</font></span></a>, <a class="txt" href="#E17:28_1_1"><span class="lab"><font color="Green" title="E26">A17</font></span></a>, <a class="ref" href="setfam_1.html#T10" title="SETFAM_1:th.10">SETFAM_1:10</a></span>;<br/><span class="kw">hence </span><a NAME="E22:28_1_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:28_1_1"><span class="lab"><font color="Green" title="E23">A14</font></span></a>, <a class="txt" href="#E19:28_1_1"><span class="lab"><font color="Green" title="E27">A18</font></span></a>, <a class="ref" href="tops_1.html#T38" title="TOPS_1:th.38">TOPS_1:38</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:28_1_2"/><span class="lab"><font color="Green" title="E21">A19</font></span>: 
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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> </span><br/><a NAME="E2:28_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A20</font></span>: 
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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="txt" href="#E3:28_1"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E3:28_1_2"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T117" title="FUNCT_1:th.117">FUNCT_1:117</a></span>;<br/><a NAME="E4:28_1_2"/><span class="kw">then </span>
 <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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_1"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="txt" href="#E1:28_1_2"><span class="lab"><font color="Green" title="E21">A19</font></span></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></span>;<br/><a NAME="E5:28_1_2"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="txt" href="#E2:28_1_2"><span class="lab"><font color="Green" title="E22">A20</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E6:28_1_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="ref" href="tops_2.html#D1" target="_self" title="TOPS_2:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:28_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28_1"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:28"/><span class="lab"><font color="Green" title="E16">A21</font></span>: 
<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="28_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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> &amp;  <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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> implies  <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>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:28_2"/><span class="lab"><font color="Green" title="E17">A22</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:28_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="c4">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  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>  )</span><br/>

<a NAME="E3:28_2"/><span class="lab"><font color="Green" title="E18">A23</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T5" title="PRE_TOPC:th.5">PRE_TOPC:5</a></span>;<br/>
<a NAME="E4:28_2"/>
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_2"><span class="lab"><font color="Green" title="E17">A22</font></span></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></span>;<br/>
<span class="kw">hence </span><a NAME="E5:28_2"/>
(  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>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_2"><span class="lab"><font color="Green" title="E18">A23</font></span></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:28"/><span class="lab"><font color="Green" title="E17">A24</font></span>: 
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>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E8:28"><span class="lab"><font color="Green" title="E16">A21</font></span></a>, <a class="txt" href="#E7:28"><span class="lab"><font color="Green" title="E15">A5</font></span></a>);<br/></span>
<div><span class="lab"><font color="Green" title="E18">A25</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:28_3"/><span class="lab"><font color="Green" title="E19">A26</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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> </span><br/><a NAME="E2:28_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="relat_1.html#T146" title="RELAT_1:th.146">RELAT_1:146</a></span>;<br/><span class="kw">hence </span><a NAME="E3:28_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:28"><span class="lab"><font color="Green" title="E17">A24</font></span></a>, <a class="txt" href="#E1:28_3"><span class="lab"><font color="Green" title="E19">A26</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E19">A27</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:28_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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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> </span><br/><a NAME="E2:28_4"/><span class="kw">then </span>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E3:28_4"/><span class="kw">then </span>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="relat_1.html#T146" title="RELAT_1:th.146">RELAT_1:146</a></span>;<br/><a NAME="E4:28_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A28</font></span>: 
 <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> 
 <span class="kw">by</span> <span class="lab"><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></span>;<br/><a NAME="E5:28_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T5" title="PRE_TOPC:th.5">PRE_TOPC:5</a></span>;<br/><span class="kw">hence </span><a NAME="E6:28_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28_4"><span class="lab"><font color="Green" title="E20">A28</font></span></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_5"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:28_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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font></span><br/><a NAME="E2:28_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A29</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></span>;<br/><a NAME="E3:28_5"/>
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
 ;<br/><span class="kw">hence </span><a NAME="E4:28_5"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_5"><span class="lab"><font color="Green" title="E20">A29</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E13:28"/>
 <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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:28"><span class="lab"><font color="Green" title="E18">A25</font></span></a>, <a class="txt" href="#E11:28"><span class="lab"><font color="Green" title="E19">A27</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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