<?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="firebrick">::  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="firebrick">::  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="E1:21"/><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="E2:21"/><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="firebrick">::  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="E4:21"/><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="#E2:21"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="finseq_1.html#T52" title="FINSEQ_1:th.52">FINSEQ_1:52</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="E6:21"/><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="E7:21"/><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="21_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="firebrick">::  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="E1:21_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="firebrick">::  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="firebrick">::  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="E2:21_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="firebrick">::  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="E3:21_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="E4:21_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="firebrick">::  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="21_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="#E4:21"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="tops_2.html#T2" target="_self" title="TOPS_2:th.2">Th3</a>, <a class="ref" href="relat_1.html#T111" title="RELAT_1:th.111">RELAT_1:111</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="#E4:21"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="tops_2.html#T2" target="_self" title="TOPS_2:th.2">Th3</a>, <a class="ref" href="relat_1.html#T111" title="RELAT_1:th.111">RELAT_1:111</a></span>;<br/><span class="kw">assume </span><a NAME="E3:21_1_1"/><span class="lab"><font color="Green" title="E21">A11</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="firebrick">::  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="E4:21_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="#E3:21_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="E5:21_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#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E6:21_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="#E6:21"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></span>;<br/><a NAME="E7:21_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A12</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="#E3:21_1_1"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="ref" href="relat_1.html#T119" title="RELAT_1:th.119">RELAT_1:119</a></span>;<br/><a NAME="E8:21_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="#E3:21_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="E9:21_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#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E10:21_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A13</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="#E1:21_1"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="txt" href="#E4:21_1"><span class="lab"><font color="Green" title="E19">A9</font></span></a></span>;<br/><a NAME="E11:21_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="E12:21_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#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/><a NAME="E13:21_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A14</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="#E6:21"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="txt" href="#E3:21_1"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E14:21_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#T59" title="FUNCT_1:th.59">FUNCT_1:59</a></span>;<br/><a NAME="E15:21_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A15</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#T10" title="SETFAM_1:th.10">SETFAM_1:10</a></span>;<br/><a NAME="E16:21_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="#E13:21_1_1"><span class="lab"><font color="Green" title="E24">A14</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/><a NAME="E17:21_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A16</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#T119" title="RELAT_1:th.119">RELAT_1:119</a></span>;<br/><a NAME="E18:21_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="#E4:21"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="txt" href="#E13:21_1_1"><span class="lab"><font color="Green" title="E24">A14</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E19:21_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A17</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="#E1:21"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E15:21_1_1"><span class="lab"><font color="Green" title="E25">A15</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#T9" title="FINSEQ_1:th.9">FINSEQ_1:9</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#T120" title="RELAT_1:th.120">RELAT_1:120</a></span>
;<br/><a NAME="E21:21_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="#E2:21_1"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="txt" href="#E7:21_1_1"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E17:21_1_1"><span class="lab"><font color="Green" title="E26">A16</font></span></a>, <a class="ref" href="setfam_1.html#T9" title="SETFAM_1:th.9">SETFAM_1:9</a></span>;<br/><span class="kw">hence </span><a NAME="E22:21_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="#E10:21_1_1"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="txt" href="#E19:21_1_1"><span class="lab"><font color="Green" title="E27">A17</font></span></a>, <a class="ref" href="tops_1.html#T11" title="TOPS_1:th.11">TOPS_1:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:21_1_2"/><span class="lab"><font color="Green" title="E21">A18</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="firebrick">::  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="E2:21_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A19</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="#E6:21"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="txt" href="#E3:21_1"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E3:21_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#T59" title="FUNCT_1:th.59">FUNCT_1:59</a></span>;<br/><a NAME="E4:21_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="#E2:21_1"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="txt" href="#E1:21_1_2"><span class="lab"><font color="Green" title="E21">A18</font></span></a>, <a class="ref" href="finseq_1.html#T2" title="FINSEQ_1:th.2">FINSEQ_1:2</a>, <a class="ref" href="setfam_1.html#T10" title="SETFAM_1:th.10">SETFAM_1:10</a></span>;<br/><a NAME="E5:21_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="#E4:21"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="txt" href="#E2:21_1_2"><span class="lab"><font color="Green" title="E22">A19</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E6:21_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="#E1:21"><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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:21_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="#E5:21_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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:21"/><span class="lab"><font color="Green" title="E16">A20</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="21_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="firebrick">::  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="E1:21_2"/><span class="lab"><font color="Green" title="E17">A21</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="E2:21_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="firebrick">::  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="E3:21_2"/><span class="lab"><font color="Green" title="E18">A22</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#T1" title="PRE_TOPC:th.1">PRE_TOPC:1</a></span>;<br/>
<a NAME="E4:21_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="#E1:21_2"><span class="lab"><font color="Green" title="E17">A21</font></span></a>, <a class="ref" href="relat_1.html#T116" title="RELAT_1:th.116">RELAT_1:116</a>, <a class="ref" href="setfam_1.html#T1" title="SETFAM_1:th.1">SETFAM_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:21_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="#E3:21_2"><span class="lab"><font color="Green" title="E18">A22</font></span></a>, <a class="ref" href="pre_topc.html#D2" title="PRE_TOPC:def.2">PRE_TOPC:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:21"/><span class="lab"><font color="Green" title="E17">A23</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="#E8:21"><span class="lab"><font color="Green" title="E16">A20</font></span></a>, <a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E15">A5</font></span></a>);<br/></span>
<div><span class="lab"><font color="Green" title="E18">A24</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:21_3"/><span class="lab"><font color="Green" title="E19">A25</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="firebrick">::  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="E2:21_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="txt" href="#E6:21"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="relat_1.html#T113" title="RELAT_1:th.113">RELAT_1:113</a></span>;<br/><span class="kw">hence </span><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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:21"><span class="lab"><font color="Green" title="E17">A23</font></span></a>, <a class="txt" href="#E1:21_3"><span class="lab"><font color="Green" title="E19">A25</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E19">A26</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E2:21_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="E3:21_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="#E4:21"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="txt" href="#E6:21"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="relat_1.html#T113" title="RELAT_1:th.113">RELAT_1:113</a></span>;<br/><a NAME="E4:21_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A27</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#T116" title="RELAT_1:th.116">RELAT_1:116</a>, <a class="ref" href="setfam_1.html#T1" title="SETFAM_1:th.1">SETFAM_1:1</a></span>;<br/><a NAME="E5:21_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T1" title="PRE_TOPC:th.1">PRE_TOPC:1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:21_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_4"><span class="lab"><font color="Green" title="E20">A27</font></span></a>, <a class="ref" href="pre_topc.html#D2" title="PRE_TOPC:def.2">PRE_TOPC:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_5"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E2:21_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A28</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="E3:21_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="E4:21_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="#E2:21_5"><span class="lab"><font color="Green" title="E20">A28</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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E13:21"/>
 <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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:21"><span class="lab"><font color="Green" title="E18">A24</font></span></a>, <a class="txt" href="#E11:21"><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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
