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

<b>set </b><font color="Maroon" title="c56">D1</font> =  <a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1;<br/>
<b>set </b><font color="Maroon" title="c57">D2</font> =  <a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>;<br/>
<a NAME="E1:117"/><i><font color="Green" title="E101">A1</font></i>: 
 <a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c58">F1</font> = <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c59">F2</font> = <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c60">F</font> = <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p4">(<span class="default"><a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p4">(<span class="default"><a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>;<br/>
<b>reconsider </b><font color="Maroon" title="c61">F</font> = <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p4">(<span class="default"><a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p4">(<span class="default"><a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p5">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> ;<br/>
<b>take </b>
<font color="Maroon" title="c61">F</font>
;<br/>

<a NAME="E3:117"/><i><font color="Green" title="E102">A2</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1
 
<b>by </b><i><a class="txt" href="toprealb.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<a NAME="E4:117"/><b>then </b><i><font color="Green" title="E103">A3</font></i>: 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="ref" href="toprealb.html#T41" target="_self" title="TOPREALB:th.41">Th41</a></i>;<br/>
<a NAME="E5:117"/>
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="toprealb.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<a NAME="E6:117"/><b>then </b><i><font color="Green" title="E104">A4</font></i>: 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="ref" href="toprealb.html#T41" target="_self" title="TOPREALB:th.41">Th41</a></i>;<br/>
<b>hence </b><a NAME="E7:117"/>
<font color="Maroon" title="c61">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span>,<span class="p2">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <b>by </b><i><a class="txt" href="toprealb.html#E3:117"><i><font color="Green" title="E102">A2</font></i></a>, <a class="ref" href="toprealb.html#T41" target="_self" title="TOPREALB:th.41">Th41</a></i>;<br/>

<b>thus </b><a NAME="E8:117"/>
<font color="Maroon" title="c61">F</font> <a href="pre_topc.html#R1" title="PRE_TOPC:pred.1">is_a_cover_of</a>  <a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:117_1"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c61">F</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>,<a class="ref" href="pre_topc.html#D8" title="PRE_TOPC:def.8">PRE_TOPC:def 8</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c62">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:117_1_1"/><i><font color="Green" title="E105">A5</font></i>: 
<font color="Maroon" title="c62">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c63">A</font> = <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c64">f</font> = <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c63">A</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c63">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <b>by </b><i><a class="txt" href="toprealb.html#E74"><i><font color="Green" title="E51">Lm23</font></i></a></i>;<br/>
<a NAME="E4:117_1_1"/>
<font color="Maroon" title="c64">f</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 
<b>by </b><i><a class="ref" href="toprealb.html#T39" target="_self" title="TOPREALB:th.39">Th39</a></i>;<br/>
<a NAME="E5:117_1_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c64">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c65">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:117_1_1"/><i><font color="Green" title="E106">A6</font></i>: 
<font color="Maroon" title="c65">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c64">f</font>
 <b>and </b><br/><a NAME="E8:117_1_1"/><i><font color="Green" title="E107">A7</font></i>: 
<font color="Maroon" title="c64">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c65">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c62">a</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E1:117_1_1"><i><font color="Green" title="E105">A5</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E9:117_1_1"/><i><font color="Green" title="E108">A8</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c64">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c63">A</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E63"><i><font color="Green" title="E43">Lm19</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c66">x</font> = <font color="Maroon" title="c65">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="toprealb.html#E7:117_1_1"><i><font color="Green" title="E106">A6</font></i></a></i>;<br/>
<a NAME="E11:117_1_1"/><i><font color="Green" title="E109">A9</font></i>: 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c66">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c64">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c66">x</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E7:117_1_1"><i><font color="Green" title="E106">A6</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<a NAME="E12:117_1_1"/><i><font color="Green" title="E110">A10</font></i>: 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c61">F</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:117_1_1_1"/>
( <font color="Maroon" title="c66">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or (  <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="c66">x</font> &amp; <font color="Maroon" title="c66">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 ) or ( <font color="Maroon" title="c66">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c66">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="toprealb.html#E7:117_1_1"><i><font color="Green" title="E106">A6</font></i></a>, <a class="txt" href="toprealb.html#E9:117_1_1"><i><font color="Green" title="E108">A8</font></i></a>, <a class="ref" href="rcomp_2.html#T3" title="RCOMP_2:th.3">RCOMP_2:3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1_1_1_1"><b>suppose </b></a><a NAME="E1:117_1_1_1_1"/><i><font color="Green" title="E111">A11</font></i>: 
<font color="Maroon" title="c66">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:117_1_1_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c63">A</font>
 <b>by </b><i><a class="ref" href="rcomp_2.html#T3" title="RCOMP_2:th.3">RCOMP_2:3</a></i>;<br/><b>then </b><i><font color="Green" title="E112">A12</font></i>: <font color="Maroon" title="c64">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a> 
<b>by </b><i><a class="ref" href="toprealb.html#T33" target="_self" title="TOPREALB:th.33">Th33</a></i>
<br/>.= 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="ref" href="toprealb.html#T33" target="_self" title="TOPREALB:th.33">Th33</a></i>
;<br/><a NAME="E4:117_1_1_1_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/><a NAME="E5:117_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c62">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="toprealb.html#E8:117_1_1"><i><font color="Green" title="E107">A7</font></i></a>, <a class="txt" href="toprealb.html#E1:117_1_1_1_1"><i><font color="Green" title="E111">A11</font></i></a>, <a class="txt" href="toprealb.html#E3:117_1_1_1_1"><i><font color="Green" title="E112">A12</font></i></a>, <a class="txt" href="toprealb.html#E63"><i><font color="Green" title="E43">Lm19</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><b>hence </b><a NAME="E6:117_1_1_1_1"/>
<font color="Maroon" title="c62">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c61">F</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E6:117"><i><font color="Green" title="E104">A4</font></i></a>, <a class="txt" href="toprealb.html#E12:117_1_1"><i><font color="Green" title="E110">A10</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1_1_1_2"><b>suppose </b></a><a NAME="E1:117_1_1_1_2"/>
(  <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="c66">x</font> &amp; <font color="Maroon" title="c66">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 ;<br/><div class="add"><a NAME="E2:117_1_1_1_2"/><b>then </b><i><font color="Green" title="E111">A13</font></i>: 
<font color="Maroon" title="c66">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/><a NAME="E3:117_1_1_1_2"/>
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1
 <b>by </b><i><a class="txt" href="toprealb.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><a NAME="E4:117_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c66">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>
 <b>by </b><i><a class="txt" href="toprealb.html#E2:117_1_1_1_2"><i><font color="Green" title="E111">A13</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><a NAME="E5:117_1_1_1_2"/><b>then </b><i><font color="Green" title="E112">A14</font></i>: 
<font color="Maroon" title="c62">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="toprealb.html#E8:117_1_1"><i><font color="Green" title="E107">A7</font></i></a>, <a class="txt" href="toprealb.html#E11:117_1_1"><i><font color="Green" title="E109">A9</font></i></a>, <a class="txt" href="toprealb.html#E63"><i><font color="Green" title="E43">Lm19</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E6:117_1_1_1_2"/>
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c61">F</font>
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E7:117_1_1_1_2"/>
<font color="Maroon" title="c62">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c61">F</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E5:117_1_1_1_2"><i><font color="Green" title="E112">A14</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_1_1_1_3"><b>suppose </b></a><a NAME="E1:117_1_1_1_3"/>
( <font color="Maroon" title="c66">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c66">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:117_1_1_1_3"/>
<font color="Maroon" title="c62">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c61">F</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>thus </b><a NAME="E2:117_1"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c61">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E9:117"/>
<font color="Maroon" title="c61">F</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c62">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a><br/>

<a NAME="E1:117_2"/><i><font color="Green" title="E105">A15</font></i>: 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K13" title="TOPREALB:func.13">CircleMap</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T162" title="RELAT_1:th.162">RELAT_1:162</a></i>;<br/>
<a NAME="E2:117_2"/><i><font color="Green" title="E106">A16</font></i>: 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K13" title="TOPREALB:func.13">CircleMap</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T162" title="RELAT_1:th.162">RELAT_1:162</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c63">r</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<div><i><font color="Green" title="E107">A17</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="117_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c64">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><a NAME="E1:117_2_1"/>
<font color="Maroon" title="c64">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c64">A</font>
 ;<br/><b>hence </b><a NAME="E2:117_2_1"/>
<font color="Maroon" title="c64">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <font color="Maroon" title="c64">A</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c64">M</font> = <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c63">r</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c63">r</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c65">N</font> = <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c63">r</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c63">r</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="toprealb.html#E4:117_2"><i><font color="Green" title="E107">A17</font></i></a></i>;<br/>
<div><i><font color="Green" title="E108">A18</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="117_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c66">A</font> be  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>reconsider </b><font color="Maroon" title="c67">B</font> = <font color="Maroon" title="c66">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <font color="Maroon" title="c66">A</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="toprealb.html#E4:117_2"><i><font color="Green" title="E107">A17</font></i></a></i>;<br/><a NAME="E2:117_2_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <font color="Maroon" title="c66">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c66">A</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><a NAME="E3:117_2_2"/><b>then </b>
<font color="Maroon" title="c67">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p3">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <font color="Maroon" title="c66">A</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <font color="Maroon" title="c66">A</font></span>)</span>
 ;<br/><b>hence </b><a NAME="E4:117_2_2"/>
<font color="Maroon" title="c66">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <font color="Maroon" title="c66">A</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tops_2.html#T32" title="TOPS_2:th.32">TOPS_2:32</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:117_2"/><b>then </b>
<font color="Maroon" title="c64">M</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
;<br/>
<a NAME="E9:117_2"/><b>then </b><i><font color="Green" title="E109">A19</font></i>: 
<span class="p1">(<span class="default"><a href="toprealb.html#K13" title="TOPREALB:func.13">CircleMap</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> <font color="Maroon" title="c63">r</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c64">M</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="t_0topsp.html#D2" title="T_0TOPSP:def.2">T_0TOPSP:def 2</a></i>;<br/>
<a NAME="E10:117_2"/>
<font color="Maroon" title="c65">N</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="toprealb.html#E7:117_2"><i><font color="Green" title="E108">A18</font></i></a></i>;<br/>
<a NAME="E11:117_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="toprealb.html#K13" title="TOPREALB:func.13">CircleMap</a> <span class="p2">(<span class="default"><a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c63">r</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c65">N</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="t_0topsp.html#D2" title="T_0TOPSP:def.2">T_0TOPSP:def 2</a></i>;<br/>
<a NAME="E12:117_2"/><b>then </b>
( <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 
<b>by </b><i><a class="txt" href="toprealb.html#E4:117"><i><font color="Green" title="E103">A3</font></i></a>, <a class="txt" href="toprealb.html#E6:117"><i><font color="Green" title="E104">A4</font></i></a>, <a class="txt" href="toprealb.html#E1:117_2"><i><font color="Green" title="E105">A15</font></i></a>, <a class="txt" href="toprealb.html#E2:117_2"><i><font color="Green" title="E106">A16</font></i></a>, <a class="txt" href="toprealb.html#E9:117_2"><i><font color="Green" title="E109">A19</font></i></a>, <a class="ref" href="tsep_1.html#T17" title="TSEP_1:th.17">TSEP_1:17</a></i>;<br/>
<b>hence </b><a NAME="E13:117_2"/>
(  not <font color="Maroon" title="c62">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c61">F</font> or <font color="Maroon" title="c62">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c62">U</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span>;<br/>

<b>thus </b><a NAME="E10:117"/>
( <font color="Maroon" title="c62">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> implies (  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c62">U</font> &amp; ( for <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>   st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 holds <br/>for <font color="Olive" title="b2">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Olive" title="b1">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c62">U</font></span>)</span>  st <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">d</font> holds <br/><font color="Olive" title="b2">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:117_3"/><i><font color="Green" title="E105">A20</font></i>: 
<font color="Maroon" title="c62">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 ;<br/>

<a NAME="E2:117_3"/>
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1
 
<b>by </b><i><a class="txt" href="toprealb.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<a NAME="E3:117_3"/><b>then </b><i><font color="Green" title="E106">A21</font></i>: 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="toprealb.html#T41" target="_self" title="TOPREALB:th.41">Th41</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c63">x1</font>, <font color="Maroon" title="c64">x2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a> ;<br/><b>assume </b><a NAME="E1:117_3_1"/>
<font color="Maroon" title="c63">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c65">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:117_3_1"/><i><font color="Green" title="E107">A22</font></i>: 
<font color="Maroon" title="c63">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c65">Z</font>
 <b>and </b><br/><a NAME="E4:117_3_1"/><i><font color="Green" title="E108">A23</font></i>: 
<font color="Maroon" title="c65">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c66">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <b> such that </b><br/><a NAME="E6:117_3_1"/><i><font color="Green" title="E109">A24</font></i>: 
<font color="Maroon" title="c65">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c66">n</font></span>)</span>,<span class="p2">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c66">n</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="toprealb.html#E4:117_3_1"><i><font color="Green" title="E108">A23</font></i></a></i>;<br/><b>assume </b><a NAME="E7:117_3_1"/><i><font color="Green" title="E110">A25</font></i>: 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c63">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c64">x2</font>
 ;<br/><b>set </b><font color="Maroon" title="c67">k</font> = <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>;<br/><b>set </b><font color="Maroon" title="c68">K</font> = <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span>,<span class="p2">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>;<br/><a NAME="E8:117_3_1"/>
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/><a NAME="E9:117_3_1"/><b>then </b><i><font color="Green" title="E111">A26</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span>,<span class="p2">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1
 ;<br/><a NAME="E10:117_3_1"/><i><font color="Green" title="E112">A27</font></i>: 
<font color="Maroon" title="c66">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c63">x1</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E3:117_3_1"><i><font color="Green" title="E107">A22</font></i></a>, <a class="txt" href="toprealb.html#E6:117_3_1"><i><font color="Green" title="E109">A24</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/><a NAME="E11:117_3_1"/>
<font color="Maroon" title="c63">x1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c66">n</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E3:117_3_1"><i><font color="Green" title="E107">A22</font></i></a>, <a class="txt" href="toprealb.html#E6:117_3_1"><i><font color="Green" title="E109">A24</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/><a NAME="E12:117_3_1"/><b>then </b><i><font color="Green" title="E113">A28</font></i>: 
<font color="Maroon" title="c63">x1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c66">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E13:117_3_1"/><b>then </b><i><font color="Green" title="E114">A29</font></i>: 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c63">x1</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c66">n</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E10:117_3_1"><i><font color="Green" title="E112">A27</font></i></a>, <a class="ref" href="int_1.html#D4" title="INT_1:def.4">INT_1:def 4</a></i>;<br/><a NAME="E14:117_3_1"/><i><font color="Green" title="E115">A30</font></i>: 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c64">x2</font>
 <b>by </b><i><a class="ref" href="int_1.html#D4" title="INT_1:def.4">INT_1:def 4</a></i>;<br/><a NAME="E15:117_3_1"/>
not <font color="Maroon" title="c63">x1</font> is <a href="int_1.html#V1" title="INT_1:attr.1">integer</a>
 <b>by </b><i><a class="txt" href="toprealb.html#E10:117_3_1"><i><font color="Green" title="E112">A27</font></i></a>, <a class="txt" href="toprealb.html#E13:117_3_1"><i><font color="Green" title="E114">A29</font></i></a>, <a class="ref" href="int_1.html#T48" title="INT_1:th.48">INT_1:48</a></i>;<br/><a NAME="E16:117_3_1"/><b>then </b><i><font color="Green" title="E116">A31</font></i>: 
not <font color="Maroon" title="c63">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:117_3_1_1"/>
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c64">x2</font>
 ;<br/><a NAME="E2:117_3_1_1"/><b>then </b>
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c63">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a> 
 <b>by </b><i><a class="txt" href="toprealb.html#E7:117_3_1"><i><font color="Green" title="E110">A25</font></i></a>, <a class="ref" href="toprealb.html#T33" target="_self" title="TOPREALB:th.33">Th33</a></i>;<br/><b>hence </b><a NAME="E3:117_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="toprealb.html#E1:117"><i><font color="Green" title="E101">A1</font></i></a>, <a class="txt" href="toprealb.html#E16:117_3_1"><i><font color="Green" title="E116">A31</font></i></a>, <a class="txt" href="toprealb.html#E63"><i><font color="Green" title="E43">Lm19</font></i></a>, <a class="ref" href="toprealb.html#T34" target="_self" title="TOPREALB:th.34">Th34</a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/></div><b>end;</b></div><a NAME="E18:117_3_1"/><b>then </b><i><font color="Green" title="E117">A32</font></i>: 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c64">x2</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E14:117_3_1"><i><font color="Green" title="E115">A30</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E19:117_3_1"/><i><font color="Green" title="E118">A33</font></i>: 
( <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c63">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c66">n</font></span>)</span></span>)</span> &amp; <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c64">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c64">x2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="toprealb.html#T32" target="_self" title="TOPREALB:th.32">Th32</a></i>;<br/><a NAME="E20:117_3_1"/>
 <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="c63">x1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c66">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="toprealb.html#E10:117_3_1"><i><font color="Green" title="E112">A27</font></i></a>, <a class="ref" href="real_2.html#T105" title="REAL_2:th.105">REAL_2:105</a></i>;<br/><a NAME="E21:117_3_1"/><b>then </b><i><font color="Green" title="E119">A34</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c66">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E22:117_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c66">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c66">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c66">n</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E12:117_3_1"><i><font color="Green" title="E113">A28</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E23:117_3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c66">n</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E24:117_3_1"/><b>then </b><i><font color="Green" title="E120">A35</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c66">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><a NAME="E25:117_3_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="c64">x2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="toprealb.html#E18:117_3_1"><i><font color="Green" title="E117">A32</font></i></a>, <a class="ref" href="real_2.html#T105" title="REAL_2:th.105">REAL_2:105</a></i>;<br/><a NAME="E26:117_3_1"/><b>then </b><i><font color="Green" title="E121">A36</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c64">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E27:117_3_1"/>
<font color="Maroon" title="c64">x2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <b>by </b><i><a class="ref" href="int_1.html#D4" title="INT_1:def.4">INT_1:def 4</a></i>;<br/><a NAME="E28:117_3_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c64">x2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E29:117_3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c64">x2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E30:117_3_1"/><b>then </b><i><font color="Green" title="E122">A37</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c64">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><a NAME="E31:117_3_1"/>
( <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c66">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c66">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c66">n</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c64">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c64">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p5"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c64">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p5"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 <b>by </b><i><a class="ref" href="toprealb.html#D11" target="_self" title="TOPREALB:def.11">Def11</a></i>;<br/><a NAME="E32:117_3_1"/><b>then </b>
(  <a href="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c66">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c64">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span></span>)</span> &amp;  <a href="sin_cos.html#K21" title="SIN_COS:func.21">sin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c66">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K21" title="SIN_COS:func.21">sin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c64">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="toprealb.html#E7:117_3_1"><i><font color="Green" title="E110">A25</font></i></a>, <a class="txt" href="toprealb.html#E19:117_3_1"><i><font color="Green" title="E118">A33</font></i></a>, <a class="ref" href="sppol_2.html#T1" title="SPPOL_2:th.1">SPPOL_2:1</a></i>;<br/><a NAME="E33:117_3_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c66">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c64">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="toprealb.html#E21:117_3_1"><i><font color="Green" title="E119">A34</font></i></a>, <a class="txt" href="toprealb.html#E24:117_3_1"><i><font color="Green" title="E120">A35</font></i></a>, <a class="txt" href="toprealb.html#E26:117_3_1"><i><font color="Green" title="E121">A36</font></i></a>, <a class="txt" href="toprealb.html#E30:117_3_1"><i><font color="Green" title="E122">A37</font></i></a>, <a class="ref" href="complex2.html#T12" title="COMPLEX2:th.12">COMPLEX2:12</a></i>;<br/><a NAME="E34:117_3_1"/><b>then </b>
<font color="Maroon" title="c63">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c66">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c64">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/><a NAME="E35:117_3_1"/><b>then </b><i><font color="Green" title="E123">A38</font></i>: 
<font color="Maroon" title="c64">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c66">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 ;<br/><a NAME="E36:117_3_1"/>
<font color="Maroon" title="c63">x1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c66">n</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E3:117_3_1"><i><font color="Green" title="E107">A22</font></i></a>, <a class="txt" href="toprealb.html#E6:117_3_1"><i><font color="Green" title="E109">A24</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/><a NAME="E37:117_3_1"/><b>then </b>
<font color="Maroon" title="c63">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c66">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E38:117_3_1"/><b>then </b>
<font color="Maroon" title="c64">x2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <b>by </b><i><a class="txt" href="toprealb.html#E35:117_3_1"><i><font color="Green" title="E123">A38</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E39:117_3_1"/><b>then </b>
<font color="Maroon" title="c64">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span>,<span class="p2">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="toprealb.html#E18:117_3_1"><i><font color="Green" title="E117">A32</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/><b>hence </b><a NAME="E40:117_3_1"/>
<font color="Maroon" title="c64">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>
 <b>by </b><i><a class="txt" href="toprealb.html#E9:117_3_1"><i><font color="Green" title="E111">A26</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:117_3"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c62">U</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E1:117_3"><i><font color="Green" title="E105">A20</font></i></a>, <a class="txt" href="toprealb.html#E3:117_3"><i><font color="Green" title="E106">A21</font></i></a>, <a class="ref" href="t_0topsp.html#T2" title="T_0TOPSP:th.2">T_0TOPSP:2</a></i>;<br/>

<b>let </b><font color="Maroon" title="c63">d</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a> ;<br/>

<b>assume </b><a NAME="E6:117_3"/><i><font color="Green" title="E107">A39</font></i>: 
<font color="Maroon" title="c63">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1
 ;<br/>

<b>let </b><font color="Maroon" title="c64">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c63">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c62">U</font></span>)</span>;<br/>

<b>assume </b><a NAME="E7:117_3"/><i><font color="Green" title="E108">A40</font></i>: 
<font color="Maroon" title="c64">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c63">d</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c65">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <b> such that </b><br/><a NAME="E9:117_3"/><i><font color="Green" title="E109">A41</font></i>: 
<font color="Maroon" title="c63">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c65">n</font></span>)</span>,<span class="p2">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c65">n</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="toprealb.html#E6:117_3"><i><font color="Green" title="E107">A39</font></i></a></i>;<br/>
<a NAME="E10:117_3"/>
( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c65">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c65">n</font> &amp; <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c65">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c65">n</font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c66">d1</font> = <font color="Maroon" title="c63">d</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <b>by </b><i><a class="txt" href="toprealb.html#E9:117_3"><i><font color="Green" title="E109">A41</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/>
<a NAME="E12:117_3"/>
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c67">U1</font> = <font color="Maroon" title="c62">U</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <b>by </b><i><a class="txt" href="toprealb.html#E1:117_3"><i><font color="Green" title="E105">A20</font></i></a>, <a class="txt" href="toprealb.html#E63"><i><font color="Green" title="E43">Lm19</font></i></a>, <a class="txt" href="toprealb.html#E67"><i><font color="Green" title="E47">Lm20</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c68">f1</font> = <font color="Maroon" title="c64">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c66">d1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c67">U1</font></span>)</span> ;<br/>
<a NAME="E15:117_3"/><i><font color="Green" title="E110">A42</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c62">U</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c62">U</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E16:117_3"/><i><font color="Green" title="E111">A43</font></i>: 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c63">d</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E6:117_3"><i><font color="Green" title="E107">A39</font></i></a>, <a class="ref" href="toprealb.html#T41" target="_self" title="TOPREALB:th.41">Th41</a></i>;<br/>
<a NAME="E17:117_3"/><i><font color="Green" title="E112">A44</font></i>: 
<font color="Maroon" title="c64">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="toprealb.html#E6:117_3"><i><font color="Green" title="E107">A39</font></i></a>, <a class="txt" href="toprealb.html#E7:117_3"><i><font color="Green" title="E108">A40</font></i></a>, <a class="txt" href="toprealb.html#E5"><i><font color="Green" title="E3">Lm3</font></i></a>, <a class="ref" href="toprealb.html#T40" target="_self" title="TOPREALB:th.40">Th40</a></i>;<br/>
<a NAME="E18:117_3"/><i><font color="Green" title="E113">A45</font></i>: 
<font color="Maroon" title="c64">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="toprealb.html#E7:117_3"><i><font color="Green" title="E108">A40</font></i></a>, <a class="ref" href="topreala.html#T29" title="TOPREALA:th.29">TOPREALA:29</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c69">J</font> = <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c65">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c65">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">p1</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c70">F</font> = <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c69">J</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c69">J</font></span>)</span>,<span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <b>by </b><i><a class="txt" href="toprealb.html#E74"><i><font color="Green" title="E51">Lm23</font></i></a></i>;<br/>
<a NAME="E21:117_3"/>
<font color="Maroon" title="c63">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c69">J</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E9:117_3"><i><font color="Green" title="E109">A41</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c71">d2</font> = <font color="Maroon" title="c63">d</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c69">J</font></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E23:117_3"/><i><font color="Green" title="E114">A46</font></i>: 
<span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c69">J</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c71">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c63">d</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E9:117_3"><i><font color="Green" title="E109">A41</font></i></a>, <a class="ref" href="pre_topc.html#T28" title="PRE_TOPC:th.28">PRE_TOPC:28</a></i>;<br/>
<a NAME="E24:117_3"/><i><font color="Green" title="E115">A47</font></i>: 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c63">d</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c69">J</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c66">d1</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E9:117_3"><i><font color="Green" title="E109">A41</font></i></a>, <a class="ref" href="relat_1.html#T103" title="RELAT_1:th.103">RELAT_1:103</a></i>;<br/>
<a NAME="E25:117_3"/><i><font color="Green" title="E116">A48</font></i>: 
<font color="Maroon" title="c68">f1</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_3_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:117_3_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c68">f1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c67">U1</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>,<a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a><br/>

<b>let </b><font color="Maroon" title="c72">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:117_3_2"/>
<font color="Maroon" title="c72">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c67">U1</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c73">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a> <b> such that </b><br/><a NAME="E4:117_3_2"/><i><font color="Green" title="E117">A49</font></i>: 
<font color="Maroon" title="c73">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c63">d</font>
 <b>and </b><br/><a NAME="E5:117_3_2"/><i><font color="Green" title="E118">A50</font></i>: 
<font color="Maroon" title="c72">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c73">a</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E1:117_3"><i><font color="Green" title="E105">A20</font></i></a>, <a class="txt" href="toprealb.html#E3:117_3"><i><font color="Green" title="E106">A21</font></i></a>, <a class="txt" href="toprealb.html#E15:117_3"><i><font color="Green" title="E110">A42</font></i></a>, <a class="txt" href="toprealb.html#E16:117_3"><i><font color="Green" title="E111">A43</font></i></a>, <a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<a NAME="E6:117_3_2"/><i><font color="Green" title="E119">A51</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c63">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c63">d</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E63"><i><font color="Green" title="E43">Lm19</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E7:117_3_2"/>
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c63">d</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c73">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c73">a</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E4:117_3_2"><i><font color="Green" title="E117">A49</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><a NAME="E8:117_3_2"/>
<font color="Maroon" title="c72">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c68">f1</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E7:117_3"><i><font color="Green" title="E108">A40</font></i></a>, <a class="txt" href="toprealb.html#E4:117_3_2"><i><font color="Green" title="E117">A49</font></i></a>, <a class="txt" href="toprealb.html#E5:117_3_2"><i><font color="Green" title="E118">A50</font></i></a>, <a class="txt" href="toprealb.html#E6:117_3_2"><i><font color="Green" title="E119">A51</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c72">nn</font> = <font color="Maroon" title="c65">n</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a>, <a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E27:117_3"/>
 <a href="toprealb.html#K13" title="TOPREALB:func.13">CircleMap</a> <span class="p1">(<span class="default"><a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> <font color="Maroon" title="c65">n</font></span>)</span> is <a href="t_0topsp.html#V1" title="T_0TOPSP:attr.1">open</a>
 
;<br/>
<a NAME="E28:117_3"/><b>then </b><i><font color="Green" title="E117">A52</font></i>: 
<font color="Maroon" title="c70">F</font> is <a href="t_0topsp.html#V1" title="T_0TOPSP:attr.1">open</a>
 
<b>by </b><i><a class="ref" href="topreala.html#T33" title="TOPREALA:th.33">TOPREALA:33</a></i>;<br/>
<a NAME="E29:117_3"/>
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c65">n</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c65">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
;<br/>
<a NAME="E30:117_3"/><b>then </b>
<font color="Maroon" title="c64">f</font> is <a href="t_0topsp.html#V1" title="T_0TOPSP:attr.1">open</a>
 
<b>by </b><i><a class="txt" href="toprealb.html#E7:117_3"><i><font color="Green" title="E108">A40</font></i></a>, <a class="txt" href="toprealb.html#E23:117_3"><i><font color="Green" title="E114">A46</font></i></a>, <a class="txt" href="toprealb.html#E24:117_3"><i><font color="Green" title="E115">A47</font></i></a>, <a class="txt" href="toprealb.html#E25:117_3"><i><font color="Green" title="E116">A48</font></i></a>, <a class="txt" href="toprealb.html#E28:117_3"><i><font color="Green" title="E117">A52</font></i></a>, <a class="ref" href="topreala.html#T31" title="TOPREALA:th.31">TOPREALA:31</a></i>;<br/>
<b>hence </b><a NAME="E31:117_3"/>
<font color="Maroon" title="c64">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 <b>by </b><i><a class="txt" href="toprealb.html#E17:117_3"><i><font color="Green" title="E112">A44</font></i></a>, <a class="txt" href="toprealb.html#E18:117_3"><i><font color="Green" title="E113">A45</font></i></a>, <a class="txt" href="toprealb.html#E25:117_3"><i><font color="Green" title="E116">A48</font></i></a>, <a class="ref" href="topreala.html#T37" title="TOPREALA:th.37">TOPREALA:37</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E11:117"/><i><font color="Green" title="E105">A53</font></i>: 
<font color="Maroon" title="c62">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c63">x1</font>, <font color="Maroon" title="c64">x2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a> ;<br/><b>assume </b><a NAME="E1:117_4"/>
<font color="Maroon" title="c63">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c65">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:117_4"/><i><font color="Green" title="E106">A54</font></i>: 
<font color="Maroon" title="c63">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c65">Z</font>
 <b>and </b><br/><a NAME="E4:117_4"/><i><font color="Green" title="E107">A55</font></i>: 
<font color="Maroon" title="c65">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c66">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <b> such that </b><br/><a NAME="E6:117_4"/><i><font color="Green" title="E108">A56</font></i>: 
<font color="Maroon" title="c65">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c66">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c66">n</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="toprealb.html#E4:117_4"><i><font color="Green" title="E107">A55</font></i></a></i>;<br/><b>assume </b><a NAME="E7:117_4"/><i><font color="Green" title="E109">A57</font></i>: 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c63">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c64">x2</font>
 ;<br/><a NAME="E8:117_4"/><i><font color="Green" title="E110">A58</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c66">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c63">x1</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E3:117_4"><i><font color="Green" title="E106">A54</font></i></a>, <a class="txt" href="toprealb.html#E6:117_4"><i><font color="Green" title="E108">A56</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/><a NAME="E9:117_4"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c66">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c66">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E10:117_4"/><b>then </b><i><font color="Green" title="E111">A59</font></i>: 
<font color="Maroon" title="c66">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c63">x1</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E8:117_4"><i><font color="Green" title="E110">A58</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E11:117_4"/><i><font color="Green" title="E112">A60</font></i>: 
<font color="Maroon" title="c63">x1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c66">n</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E3:117_4"><i><font color="Green" title="E106">A54</font></i></a>, <a class="txt" href="toprealb.html#E6:117_4"><i><font color="Green" title="E108">A56</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/><a NAME="E12:117_4"/><b>then </b><i><font color="Green" title="E113">A61</font></i>: 
<font color="Maroon" title="c63">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c66">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><b>set </b><font color="Maroon" title="c67">k</font> = <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>;<br/><a NAME="E13:117_4"/><i><font color="Green" title="E114">A62</font></i>: 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c64">x2</font>
 <b>by </b><i><a class="ref" href="int_1.html#D4" title="INT_1:def.4">INT_1:def 4</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:117_4_1"/>
( <font color="Maroon" title="c63">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c66">n</font> or <font color="Maroon" title="c63">x1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c66">n</font> or <font color="Maroon" title="c63">x1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c66">n</font> )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_4_1_1"><b>suppose </b></a><a NAME="E1:117_4_1_1"/>
<font color="Maroon" title="c63">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c66">n</font>
 ;<br/><div class="add"><a NAME="E2:117_4_1_1"/><b>then </b>
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c64">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a> 
 <b>by </b><i><a class="txt" href="toprealb.html#E7:117_4"><i><font color="Green" title="E109">A57</font></i></a>, <a class="ref" href="toprealb.html#T33" target="_self" title="TOPREALB:th.33">Th33</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c68">w</font> = <font color="Maroon" title="c64">x2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <b>by </b><i><a class="txt" href="toprealb.html#E1:117"><i><font color="Green" title="E101">A1</font></i></a>, <a class="txt" href="toprealb.html#E63"><i><font color="Green" title="E43">Lm19</font></i></a>, <a class="ref" href="toprealb.html#T34" target="_self" title="TOPREALB:th.34">Th34</a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/><a NAME="E4:117_4_1_1"/>
<font color="Maroon" title="c68">w</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/><a NAME="E5:117_4_1_1"/><b>then </b><i><font color="Green" title="E115">A63</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c68">w</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c68">w</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 ;<br/><a NAME="E6:117_4_1_1"/><i><font color="Green" title="E116">A64</font></i>: 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c68">w</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c68">w</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E7:117_4_1_1"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c68">w</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c68">w</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E8:117_4_1_1"/><b>then </b>
<font color="Maroon" title="c64">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c68">w</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c68">w</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="toprealb.html#E6:117_4_1_1"><i><font color="Green" title="E116">A64</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/><b>hence </b><a NAME="E9:117_4_1_1"/>
<font color="Maroon" title="c64">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="toprealb.html#E5:117_4_1_1"><i><font color="Green" title="E115">A63</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_4_1_2"><b>suppose </b></a><a NAME="E1:117_4_1_2"/><i><font color="Green" title="E115">A65</font></i>: 
<font color="Maroon" title="c63">x1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c66">n</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c68">K</font> = <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>;<br/><a NAME="E2:117_4_1_2"/>
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/><a NAME="E3:117_4_1_2"/><b>then </b><i><font color="Green" title="E116">A66</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 ;<br/><a NAME="E4:117_4_1_2"/><i><font color="Green" title="E117">A67</font></i>: 
( <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c63">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c66">n</font></span>)</span></span>)</span> &amp; <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c64">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c64">x2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="toprealb.html#T32" target="_self" title="TOPREALB:th.32">Th32</a></i>;<br/><a NAME="E5:117_4_1_2"/>
( <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c66">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c66">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c66">n</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c64">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c64">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p5"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c64">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p5"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 <b>by </b><i><a class="ref" href="toprealb.html#D11" target="_self" title="TOPREALB:def.11">Def11</a></i>;<br/><a NAME="E6:117_4_1_2"/><b>then </b><i><font color="Green" title="E118">A68</font></i>: 
(  <a href="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c66">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c64">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span></span>)</span> &amp;  <a href="sin_cos.html#K21" title="SIN_COS:func.21">sin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c66">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K21" title="SIN_COS:func.21">sin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c64">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="toprealb.html#E7:117_4"><i><font color="Green" title="E109">A57</font></i></a>, <a class="txt" href="toprealb.html#E4:117_4_1_2"><i><font color="Green" title="E117">A67</font></i></a>, <a class="ref" href="sppol_2.html#T1" title="SPPOL_2:th.1">SPPOL_2:1</a></i>;<br/><a NAME="E7:117_4_1_2"/>
 <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="c63">x1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c66">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="toprealb.html#E10:117_4"><i><font color="Green" title="E111">A59</font></i></a>, <a class="ref" href="real_2.html#T105" title="REAL_2:th.105">REAL_2:105</a></i>;<br/><a NAME="E8:117_4_1_2"/><b>then </b><i><font color="Green" title="E119">A69</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c66">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E9:117_4_1_2"/>
<font color="Maroon" title="c63">x1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c66">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="txt" href="toprealb.html#E1:117_4_1_2"><i><font color="Green" title="E115">A65</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E10:117_4_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c66">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c66">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c66">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E11:117_4_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c66">n</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E12:117_4_1_2"/><b>then </b><i><font color="Green" title="E120">A70</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c66">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><a NAME="E13:117_4_1_2"/>
<font color="Maroon" title="c64">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c64">x2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c64">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <b>by </b><i><a class="txt" href="toprealb.html#E13:117_4"><i><font color="Green" title="E114">A62</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E14:117_4_1_2"/><b>then </b><i><font color="Green" title="E121">A71</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c64">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E15:117_4_1_2"/>
<font color="Maroon" title="c64">x2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <b>by </b><i><a class="ref" href="int_1.html#D4" title="INT_1:def.4">INT_1:def 4</a></i>;<br/><a NAME="E16:117_4_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c64">x2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E17:117_4_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c64">x2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E18:117_4_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c64">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><a NAME="E19:117_4_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c66">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c64">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="toprealb.html#E6:117_4_1_2"><i><font color="Green" title="E118">A68</font></i></a>, <a class="txt" href="toprealb.html#E8:117_4_1_2"><i><font color="Green" title="E119">A69</font></i></a>, <a class="txt" href="toprealb.html#E12:117_4_1_2"><i><font color="Green" title="E120">A70</font></i></a>, <a class="txt" href="toprealb.html#E14:117_4_1_2"><i><font color="Green" title="E121">A71</font></i></a>, <a class="ref" href="complex2.html#T12" title="COMPLEX2:th.12">COMPLEX2:12</a></i>;<br/><a NAME="E20:117_4_1_2"/><b>then </b><i><font color="Green" title="E122">A72</font></i>: 
<font color="Maroon" title="c63">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c66">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c64">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/><a NAME="E21:117_4_1_2"/><b>then </b><i><font color="Green" title="E123">A73</font></i>: 
<font color="Maroon" title="c64">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c66">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 ;<br/><a NAME="E22:117_4_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c66">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c66">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c63">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c66">n</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E8:117_4"><i><font color="Green" title="E110">A58</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E23:117_4_1_2"/><b>then </b><i><font color="Green" title="E124">A74</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c66">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E24:117_4_1_2"/>
<font color="Maroon" title="c64">x2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <b>by </b><i><a class="txt" href="toprealb.html#E12:117_4"><i><font color="Green" title="E113">A61</font></i></a>, <a class="txt" href="toprealb.html#E21:117_4_1_2"><i><font color="Green" title="E123">A73</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E25:117_4_1_2"/><b>then </b>
<font color="Maroon" title="c64">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="toprealb.html#E20:117_4_1_2"><i><font color="Green" title="E122">A72</font></i></a>, <a class="txt" href="toprealb.html#E23:117_4_1_2"><i><font color="Green" title="E124">A74</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/><b>hence </b><a NAME="E26:117_4_1_2"/>
<font color="Maroon" title="c64">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="toprealb.html#E3:117_4_1_2"><i><font color="Green" title="E116">A66</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="117_4_1_3"><b>suppose </b></a><a NAME="E1:117_4_1_3"/><i><font color="Green" title="E115">A75</font></i>: 
<font color="Maroon" title="c63">x1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c66">n</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c68">K</font> = <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>;<br/><a NAME="E2:117_4_1_3"/>
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/><a NAME="E3:117_4_1_3"/><b>then </b><i><font color="Green" title="E116">A76</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 ;<br/><a NAME="E4:117_4_1_3"/><i><font color="Green" title="E117">A77</font></i>: 
( <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c63">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c66">n</font></span>)</span></span>)</span> &amp; <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c64">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c64">x2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p3"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="toprealb.html#T32" target="_self" title="TOPREALB:th.32">Th32</a></i>;<br/><a NAME="E5:117_4_1_3"/>
( <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c66">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c66">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c66">n</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c64">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c64">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p5"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="sin_cos.html#K21" title="SIN_COS:func.21">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c64">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p5"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 <b>by </b><i><a class="ref" href="toprealb.html#D11" target="_self" title="TOPREALB:def.11">Def11</a></i>;<br/><a NAME="E6:117_4_1_3"/><b>then </b><i><font color="Green" title="E118">A78</font></i>: 
(  <a href="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c66">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K24" title="SIN_COS:func.24">cos</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c64">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span></span>)</span> &amp;  <a href="sin_cos.html#K21" title="SIN_COS:func.21">sin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c66">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K21" title="SIN_COS:func.21">sin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c64">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="toprealb.html#E7:117_4"><i><font color="Green" title="E109">A57</font></i></a>, <a class="txt" href="toprealb.html#E4:117_4_1_3"><i><font color="Green" title="E117">A77</font></i></a>, <a class="ref" href="sppol_2.html#T1" title="SPPOL_2:th.1">SPPOL_2:1</a></i>;<br/><a NAME="E7:117_4_1_3"/><i><font color="Green" title="E119">A79</font></i>: 
<font color="Maroon" title="c63">x1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c66">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="txt" href="toprealb.html#E11:117_4"><i><font color="Green" title="E112">A60</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E8:117_4_1_3"/><i><font color="Green" title="E120">A80</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c66">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c63">x1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="txt" href="toprealb.html#E1:117_4_1_3"><i><font color="Green" title="E115">A75</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E9:117_4_1_3"/><b>then </b>
<font color="Maroon" title="c66">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c66">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c66">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E10:117_4_1_3"/><b>then </b><i><font color="Green" title="E121">A81</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c66">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E11:117_4_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c66">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c66">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c66">n</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E7:117_4_1_3"><i><font color="Green" title="E119">A79</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E12:117_4_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c66">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E13:117_4_1_3"/><b>then </b><i><font color="Green" title="E122">A82</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c66">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><a NAME="E14:117_4_1_3"/>
<font color="Maroon" title="c64">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c64">x2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c64">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <b>by </b><i><a class="txt" href="toprealb.html#E13:117_4"><i><font color="Green" title="E114">A62</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E15:117_4_1_3"/><b>then </b><i><font color="Green" title="E123">A83</font></i>: 
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c64">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/><a NAME="E16:117_4_1_3"/>
<font color="Maroon" title="c64">x2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <b>by </b><i><a class="ref" href="int_1.html#D4" title="INT_1:def.4">INT_1:def 4</a></i>;<br/><a NAME="E17:117_4_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c64">x2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E18:117_4_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c64">x2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E19:117_4_1_3"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c64">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><a NAME="E20:117_4_1_3"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c66">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c64">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="toprealb.html#E6:117_4_1_3"><i><font color="Green" title="E118">A78</font></i></a>, <a class="txt" href="toprealb.html#E10:117_4_1_3"><i><font color="Green" title="E121">A81</font></i></a>, <a class="txt" href="toprealb.html#E13:117_4_1_3"><i><font color="Green" title="E122">A82</font></i></a>, <a class="txt" href="toprealb.html#E15:117_4_1_3"><i><font color="Green" title="E123">A83</font></i></a>, <a class="ref" href="complex2.html#T12" title="COMPLEX2:th.12">COMPLEX2:12</a></i>;<br/><a NAME="E21:117_4_1_3"/><b>then </b><i><font color="Green" title="E124">A84</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c66">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c64">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/><a NAME="E22:117_4_1_3"/><b>then </b><i><font color="Green" title="E125">A85</font></i>: 
<font color="Maroon" title="c64">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c66">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 ;<br/><a NAME="E23:117_4_1_3"/><i><font color="Green" title="E126">A86</font></i>: 
<font color="Maroon" title="c66">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c66">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c66">n</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E8:117_4_1_3"><i><font color="Green" title="E120">A80</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E24:117_4_1_3"/>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E25:117_4_1_3"/><b>then </b><i><font color="Green" title="E127">A87</font></i>: 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c66">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <b>by </b><i><a class="txt" href="toprealb.html#E23:117_4_1_3"><i><font color="Green" title="E126">A86</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E26:117_4_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c63">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c66">n</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <b>by </b><i><a class="txt" href="toprealb.html#E12:117_4"><i><font color="Green" title="E113">A61</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E27:117_4_1_3"/><b>then </b>
<font color="Maroon" title="c64">x2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 <b>by </b><i><a class="txt" href="toprealb.html#E22:117_4_1_3"><i><font color="Green" title="E125">A85</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E28:117_4_1_3"/><b>then </b>
<font color="Maroon" title="c64">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c64">x2</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="toprealb.html#E21:117_4_1_3"><i><font color="Green" title="E124">A84</font></i></a>, <a class="txt" href="toprealb.html#E25:117_4_1_3"><i><font color="Green" title="E127">A87</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/><b>hence </b><a NAME="E29:117_4_1_3"/>
<font color="Maroon" title="c64">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="toprealb.html#E3:117_4_1_3"><i><font color="Green" title="E116">A76</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E13:117"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c62">U</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E6:117"><i><font color="Green" title="E104">A4</font></i></a>, <a class="txt" href="toprealb.html#E11:117"><i><font color="Green" title="E105">A53</font></i></a>, <a class="ref" href="t_0topsp.html#T2" title="T_0TOPSP:th.2">T_0TOPSP:2</a></i>;<br/>

<b>let </b><font color="Maroon" title="c63">d</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a> ;<br/>

<b>assume </b><a NAME="E14:117"/><i><font color="Green" title="E106">A88</font></i>: 
<font color="Maroon" title="c63">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>
 ;<br/>

<b>let </b><font color="Maroon" title="c64">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c63">d</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c62">U</font></span>)</span>;<br/>

<b>assume </b><a NAME="E15:117"/><i><font color="Green" title="E107">A89</font></i>: 
<font color="Maroon" title="c64">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c63">d</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c65">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <b> such that </b><br/><a NAME="E17:117"/><i><font color="Green" title="E108">A90</font></i>: 
<font color="Maroon" title="c63">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c65">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c65">n</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="toprealb.html#E14:117"><i><font color="Green" title="E106">A88</font></i></a></i>;<br/>
<a NAME="E18:117"/>
( <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c65">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c65">n</font> &amp; 1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c65">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c65">n</font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c66">d1</font> = <font color="Maroon" title="c63">d</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <b>by </b><i><a class="txt" href="toprealb.html#E17:117"><i><font color="Green" title="E108">A90</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/>
<a NAME="E20:117"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c67">U1</font> = <font color="Maroon" title="c62">U</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <b>by </b><i><a class="txt" href="toprealb.html#E11:117"><i><font color="Green" title="E105">A53</font></i></a>, <a class="txt" href="toprealb.html#E63"><i><font color="Green" title="E43">Lm19</font></i></a>, <a class="txt" href="toprealb.html#E68"><i><font color="Green" title="E48">Lm21</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c68">f1</font> = <font color="Maroon" title="c64">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c66">d1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c67">U1</font></span>)</span> ;<br/>
<a NAME="E23:117"/><i><font color="Green" title="E109">A91</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c62">U</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c62">U</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E24:117"/><i><font color="Green" title="E110">A92</font></i>: 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="toprealb.html#K3" title="TOPREALB:func.3">IntIntervals</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c63">d</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E14:117"><i><font color="Green" title="E106">A88</font></i></a>, <a class="ref" href="toprealb.html#T41" target="_self" title="TOPREALB:th.41">Th41</a></i>;<br/>
<a NAME="E25:117"/><i><font color="Green" title="E111">A93</font></i>: 
<font color="Maroon" title="c64">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="toprealb.html#E14:117"><i><font color="Green" title="E106">A88</font></i></a>, <a class="txt" href="toprealb.html#E15:117"><i><font color="Green" title="E107">A89</font></i></a>, <a class="txt" href="toprealb.html#E6"><i><font color="Green" title="E4">Lm4</font></i></a>, <a class="ref" href="toprealb.html#T40" target="_self" title="TOPREALB:th.40">Th40</a></i>;<br/>
<a NAME="E26:117"/><i><font color="Green" title="E112">A94</font></i>: 
<font color="Maroon" title="c64">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="toprealb.html#E15:117"><i><font color="Green" title="E107">A89</font></i></a>, <a class="ref" href="topreala.html#T29" title="TOPREALA:th.29">TOPREALA:29</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c69">J</font> = <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c65">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c65">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c5">p1</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c70">F</font> = <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c69">J</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c69">J</font></span>)</span>,<span class="p1">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <b>by </b><i><a class="txt" href="toprealb.html#E74"><i><font color="Green" title="E51">Lm23</font></i></a></i>;<br/>
<a NAME="E29:117"/>
<font color="Maroon" title="c63">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c69">J</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E17:117"><i><font color="Green" title="E108">A90</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c71">d2</font> = <font color="Maroon" title="c63">d</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c69">J</font></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E31:117"/><i><font color="Green" title="E113">A95</font></i>: 
<span class="p1">(<span class="default"><a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c69">J</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c71">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c63">d</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E17:117"><i><font color="Green" title="E108">A90</font></i></a>, <a class="ref" href="pre_topc.html#T28" title="PRE_TOPC:th.28">PRE_TOPC:28</a></i>;<br/>
<a NAME="E32:117"/><i><font color="Green" title="E114">A96</font></i>: 
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c63">d</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c69">J</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c66">d1</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E17:117"><i><font color="Green" title="E108">A90</font></i></a>, <a class="ref" href="relat_1.html#T103" title="RELAT_1:th.103">RELAT_1:103</a></i>;<br/>
<a NAME="E33:117"/><i><font color="Green" title="E115">A97</font></i>: 
<font color="Maroon" title="c68">f1</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:117_5"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c68">f1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c67">U1</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>,<a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a><br/>

<b>let </b><font color="Maroon" title="c72">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:117_5"/>
<font color="Maroon" title="c72">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c67">U1</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c73">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a> <b> such that </b><br/><a NAME="E4:117_5"/><i><font color="Green" title="E116">A98</font></i>: 
<font color="Maroon" title="c73">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c63">d</font>
 <b>and </b><br/><a NAME="E5:117_5"/><i><font color="Green" title="E117">A99</font></i>: 
<font color="Maroon" title="c72">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c73">a</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E6:117"><i><font color="Green" title="E104">A4</font></i></a>, <a class="txt" href="toprealb.html#E11:117"><i><font color="Green" title="E105">A53</font></i></a>, <a class="txt" href="toprealb.html#E23:117"><i><font color="Green" title="E109">A91</font></i></a>, <a class="txt" href="toprealb.html#E24:117"><i><font color="Green" title="E110">A92</font></i></a>, <a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<a NAME="E6:117_5"/><i><font color="Green" title="E118">A100</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c63">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c63">d</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E63"><i><font color="Green" title="E43">Lm19</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E7:117_5"/>
<span class="p1">(<span class="default"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c63">d</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c73">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c73">a</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E4:117_5"><i><font color="Green" title="E116">A98</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><a NAME="E8:117_5"/>
<font color="Maroon" title="c72">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c68">f1</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E15:117"><i><font color="Green" title="E107">A89</font></i></a>, <a class="txt" href="toprealb.html#E4:117_5"><i><font color="Green" title="E116">A98</font></i></a>, <a class="txt" href="toprealb.html#E5:117_5"><i><font color="Green" title="E117">A99</font></i></a>, <a class="txt" href="toprealb.html#E6:117_5"><i><font color="Green" title="E118">A100</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c72">nn</font> = <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c65">n</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a>, <a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E35:117"/>
 <a href="toprealb.html#K13" title="TOPREALB:func.13">CircleMap</a> <span class="p1">(<span class="default"><a href="toprealb.html#K4" title="TOPREALB:func.4">R^1</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c65">n</font></span>)</span></span>)</span> is <a href="t_0topsp.html#V1" title="T_0TOPSP:attr.1">open</a>
 
;<br/>
<a NAME="E36:117"/><b>then </b><i><font color="Green" title="E116">A101</font></i>: 
<font color="Maroon" title="c70">F</font> is <a href="t_0topsp.html#V1" title="T_0TOPSP:attr.1">open</a>
 
<b>by </b><i><a class="ref" href="topreala.html#T33" title="TOPREALA:th.33">TOPREALA:33</a></i>;<br/>
<a NAME="E37:117"/>
<a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c65">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c65">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
;<br/>
<a NAME="E38:117"/><b>then </b>
<font color="Maroon" title="c64">f</font> is <a href="t_0topsp.html#V1" title="T_0TOPSP:attr.1">open</a>
 
<b>by </b><i><a class="txt" href="toprealb.html#E15:117"><i><font color="Green" title="E107">A89</font></i></a>, <a class="txt" href="toprealb.html#E31:117"><i><font color="Green" title="E113">A95</font></i></a>, <a class="txt" href="toprealb.html#E32:117"><i><font color="Green" title="E114">A96</font></i></a>, <a class="txt" href="toprealb.html#E33:117"><i><font color="Green" title="E115">A97</font></i></a>, <a class="txt" href="toprealb.html#E36:117"><i><font color="Green" title="E116">A101</font></i></a>, <a class="ref" href="topreala.html#T31" title="TOPREALA:th.31">TOPREALA:31</a></i>;<br/>
<b>hence </b><a NAME="E39:117"/>
<font color="Maroon" title="c64">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 <b>by </b><i><a class="txt" href="toprealb.html#E25:117"><i><font color="Green" title="E111">A93</font></i></a>, <a class="txt" href="toprealb.html#E26:117"><i><font color="Green" title="E112">A94</font></i></a>, <a class="txt" href="toprealb.html#E33:117"><i><font color="Green" title="E115">A97</font></i></a>, <a class="ref" href="topreala.html#T37" title="TOPREALA:th.37">TOPREALA:37</a></i>;<br/>


</div>
