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

<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="prob_1.html#K3" title="PROB_1:func.3">Intersection</a> <span class="p1">(<span class="default"><a href="finance1.html#K2" title="FINANCE1:func.2">half_open_sets</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a> </span><br/>

<a NAME="E1:21"/>
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <span class="p2">(<span class="default"><a href="finance1.html#K2" title="FINANCE1:func.2">half_open_sets</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a> 
 
<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="c3">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <span class="p2">(<span class="default"><a href="finance1.html#K2" title="FINANCE1:func.2">half_open_sets</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a> </span><br/>

<a NAME="E1:21_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finance1.html#K2" title="FINANCE1:func.2">half_open_sets</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:21_1_1"/>
<span class="p1">(<span class="default"><a href="finance1.html#K2" title="FINANCE1:func.2">half_open_sets</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:21_1_1_1_1"/>
( <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  ex <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" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:21_1_1_1_1_1"/><span class="lab"><font color="Green" title="E12">CJ00</font></span>: 
<font color="Maroon" title="c3">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"> <span class="p1">(<span class="default"><a href="finance1.html#K2" title="FINANCE1:func.2">half_open_sets</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a> </span><br/><div class="add"><a NAME="E2:21_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="finance1.html#K2" title="FINANCE1:func.2">half_open_sets</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finance1.html#K1" title="FINANCE1:func.1">halfline_fin</a> (<font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finance1.html#D1" target="_self" title="FINANCE1:def.1">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:21_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="finance1.html#K2" title="FINANCE1:func.2">half_open_sets</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1_1_1_1"><span class="lab"><font color="Green" title="E12">CJ00</font></span></a>, <a class="ref" href="finance1.html#T4" target="_self" title="FINANCE1:th.4">BJ2</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="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:21_1_1_1_1_2"/>
 ex <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" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="finance1.html#K2" title="FINANCE1:func.2">half_open_sets</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:21_1_1_1_1_2"/><span class="lab"><font color="Green" title="E12">CJ00</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">k</font> = <font color="Maroon" title="c4">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E5:21_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="finance1.html#K2" title="FINANCE1:func.2">half_open_sets</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finance1.html#K1" title="FINANCE1:func.1">halfline_fin</a> (<font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K18" title="BINOP_2:func.18">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finance1.html#D1" target="_self" title="FINANCE1:def.1">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E6:21_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="finance1.html#K2" title="FINANCE1:func.2">half_open_sets</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_1_1_1_1_2"><span class="lab"><font color="Green" title="E12">CJ00</font></span></a>, <a class="ref" href="finance1.html#T4" target="_self" title="FINANCE1:th.4">BJ2</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="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:21_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finance1.html#K2" title="FINANCE1:func.2">half_open_sets</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#D1" title="PROB_1:def.1">PROB_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:21_1"/>
<span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <span class="p2">(<span class="default"><a href="finance1.html#K2" title="FINANCE1:func.2">half_open_sets</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">n</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#D2" title="PROB_1:def.2">PROB_1: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="E2:21"/><span class="kw">then </span>
 <a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <span class="p1">(<span class="default"><a href="finance1.html#K2" title="FINANCE1:func.2">half_open_sets</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> is   <a href="prob_1.html#NM4" title="PROB_1:NM.4">SetSequence</a> of  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T25" title="PROB_1:th.25">PROB_1:25</a></span>;<br/>
<a NAME="E3:21"/><span class="kw">then </span>
 <a href="prob_1.html#K1" title="PROB_1:func.1">Union</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Complement</a> <span class="p2">(<span class="default"><a href="finance1.html#K2" title="FINANCE1:func.2">half_open_sets</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span>)</span> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#T26" title="PROB_1:th.26">PROB_1:26</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:21"/>
 <a href="prob_1.html#K3" title="PROB_1:func.3">Intersection</a> <span class="p1">(<span class="default"><a href="finance1.html#K2" title="FINANCE1:func.2">half_open_sets</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="prob_1.html#K12" title="PROB_1:func.12">Borel_Sets</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prob_1.html#D1" title="PROB_1:def.1">PROB_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
