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

<b>let </b><font color="Maroon" title="c56">aX1</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#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <a href="toprealb.html#K10" title="TOPREALB:func.10">c[-10]</a> </span>)</span>;<br/>

<b>assume </b><a NAME="E1:109"/><i><font color="Green" title="E97">A1</font></i>: 
<font color="Maroon" title="c56">aX1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">q</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"><a href="toprealb.html#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <a href="toprealb.html#K10" title="TOPREALB:func.10">c[-10]</a> </span>)</span> &amp; 0 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c57">c1</font> =  <a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a>  as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E3:109"/><i><font color="Green" title="E98">A2</font></i>: 
<font color="Maroon" title="c57">c1</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E4:109"/>
 <a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a>  is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <a href="toprealb.html#K10" title="TOPREALB:func.10">c[-10]</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="toprealb.html#E47"><i><font color="Green" title="E30">Lm16</font></i></a>, <a class="ref" href="toprealb.html#T24" target="_self" title="TOPREALB:th.24">Th24</a></i>;<br/>
<a NAME="E5:109"/><b>then </b>
 <a href="toprealb.html#K9" title="TOPREALB:func.9">c[10]</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c56">aX1</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E1:109"><i><font color="Green" title="E97">A1</font></i></a>, <a class="txt" href="toprealb.html#E3:109"><i><font color="Green" title="E98">A2</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c58">aX1</font> = <font color="Maroon" title="c56">aX1</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#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <a href="toprealb.html#K10" title="TOPREALB:func.10">c[-10]</a> </span>)</span> ;<br/>
<b>set </b><font color="Maroon" title="c59">X1</font> = <span class="p1">(<span class="default"><a href="toprealb.html#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <a href="toprealb.html#K10" title="TOPREALB:func.10">c[-10]</a> </span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c58">aX1</font>;<br/>
<a NAME="E7:109"/><i><font color="Green" title="E99">A3</font></i>: 
<font color="Maroon" title="c58">aX1</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"><span class="p2">(<span class="default"><a href="toprealb.html#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <a href="toprealb.html#K10" title="TOPREALB:func.10">c[-10]</a> </span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c58">aX1</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="E8:109"/><i><font color="Green" title="E100">A4</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#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <a href="toprealb.html#K10" title="TOPREALB:func.10">c[-10]</a> </span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c58">aX1</font></span>)</span> is   <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#E12"><i><font color="Green" title="E10">Lm10</font></i></a></i>;<br/>
<a NAME="E9:109"/>
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> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T1" title="TSEP_1:th.1">TSEP_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c60">B</font> =  <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#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <a href="toprealb.html#K10" title="TOPREALB:func.10">c[-10]</a> </span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c58">aX1</font></span>)</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 <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="toprealb.html#E8:109"><i><font color="Green" title="E100">A4</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:109"/>
<span class="p1">(<span class="default"><a href="toprealb.html#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <a href="toprealb.html#K10" title="TOPREALB:func.10">c[-10]</a> </span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c58">aX1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of  <a href="toprealb.html#K8" title="TOPREALB:func.8">Tunit_circle</a> 2
 
<b>by </b><i><a class="ref" href="tsep_1.html#T7" title="TSEP_1:th.7">TSEP_1:7</a></i>;<br/>
<a NAME="E12:109"/><b>then </b>
<span class="p1">(<span class="default"><a href="toprealb.html#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <a href="toprealb.html#K10" title="TOPREALB:func.10">c[-10]</a> </span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c58">aX1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2
 
<b>by </b><i><a class="ref" href="tsep_1.html#T7" title="TSEP_1:th.7">TSEP_1:7</a></i>;<br/>
<a NAME="E13:109"/><b>then </b><i><font color="Green" title="E101">A5</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c60">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <a href="toprealb.html#K10" title="TOPREALB:func.10">c[-10]</a> </span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c58">aX1</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/>
<b>set </b><font color="Maroon" title="c61">f</font> = <font color="Maroon" title="c38">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c60">B</font>;<br/>
<a NAME="E14:109"/><i><font color="Green" title="E102">A6</font></i>: 
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="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c60">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c60">B</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="E15:109"/><i><font color="Green" title="E103">A7</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c60">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c60">B</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E119"><i><font color="Green" title="E77">Lm41</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<a NAME="E16:109"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c38">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c60">B</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c44">Q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c62">y</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="E1:109_1"/>
<font color="Maroon" title="c62">y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c38">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c60">B</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c63">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:109_1"/><i><font color="Green" title="E104">A8</font></i>: 
<font color="Maroon" title="c63">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> <span class="p1">(<span class="default"><font color="Maroon" title="c38">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c60">B</font></span>)</span>
 <b>and </b><br/><a NAME="E4:109_1"/><i><font color="Green" title="E105">A9</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c38">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c60">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c63">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c62">y</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c64">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E6:109_1"/><i><font color="Green" title="E106">A10</font></i>: 
<font color="Maroon" title="c64">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c63">x</font>
 <b>and </b><br/><a NAME="E7:109_1"/><i><font color="Green" title="E107">A11</font></i>: 
<font color="Maroon" title="c64">q</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"><a href="toprealb.html#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <a href="toprealb.html#K10" title="TOPREALB:func.10">c[-10]</a> </span>)</span>
 <b>and </b><br/><a NAME="E8:109_1"/>
0 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c64">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="toprealb.html#E1:109"><i><font color="Green" title="E97">A1</font></i></a>, <a class="txt" href="toprealb.html#E7:109"><i><font color="Green" title="E99">A3</font></i></a>, <a class="txt" href="toprealb.html#E15:109"><i><font color="Green" title="E103">A7</font></i></a>, <a class="txt" href="toprealb.html#E3:109_1"><i><font color="Green" title="E104">A8</font></i></a></i>;<br/>
<i><font color="Green" title="E108">A12</font></i>: <font color="Maroon" title="c62">y</font> = 
<font color="Maroon" title="c38">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c63">x</font>
<b>by </b><i><a class="txt" href="toprealb.html#E15:109"><i><font color="Green" title="E103">A7</font></i></a>, <a class="txt" href="toprealb.html#E3:109_1"><i><font color="Green" title="E104">A8</font></i></a>, <a class="txt" href="toprealb.html#E4:109_1"><i><font color="Green" title="E105">A9</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c64">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
<b>by </b><i><a class="txt" href="toprealb.html#E6:109_1"><i><font color="Green" title="E106">A10</font></i></a>, <a class="ref" href="pscomp_1.html#D28" title="PSCOMP_1:def.28">PSCOMP_1:def 28</a></i>
;<br/>
<a NAME="E10:109_1"/>
(  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c64">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c64">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="toprealb.html#E7:109_1"><i><font color="Green" title="E107">A11</font></i></a>, <a class="ref" href="toprealb.html#T29" target="_self" title="TOPREALB:th.29">Th29</a></i>;<br/>
<b>hence </b><a NAME="E11:109_1"/>
<font color="Maroon" title="c62">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c44">Q</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E9:109_1"><i><font color="Green" title="E108">A12</font></i></a>, <a class="ref" href="rcomp_2.html#T4" title="RCOMP_2:th.4">RCOMP_2:4</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c62">f</font> = <font color="Maroon" title="c38">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c60">B</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c60">B</font></span>)</span>,<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="c44">Q</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="toprealb.html#E14:109"><i><font color="Green" title="E102">A6</font></i></a>, <a class="txt" href="toprealb.html#E15:109"><i><font color="Green" title="E103">A7</font></i></a>, <a class="txt" href="toprealb.html#E127"><i><font color="Green" title="E83">Lm47</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/>
<a NAME="E18:109"/>
<font color="Maroon" title="c62">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#E120"><i><font color="Green" title="E78">Lm42</font></i></a>, <a class="ref" href="topreala.html#T29" title="TOPREALA:th.29">TOPREALA:29</a></i>;<br/>
<a NAME="E19:109"/><b>then </b>
<font color="Maroon" title="c53">c</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c62">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#E155"><i><font color="Green" title="E94">Lm58</font></i></a>, <a class="ref" href="tops_2.html#T58" title="TOPS_2:th.58">TOPS_2:58</a></i>;<br/>
<a NAME="E20:109"/><b>then </b><i><font color="Green" title="E104">A13</font></i>: 
<font color="Maroon" title="c48">Af</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c53">c</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c62">f</font></span>)</span> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#T58" title="TOPS_2:th.58">TOPS_2:58</a></i>;<br/>
<a NAME="E21:109"/>
for <font color="Olive" title="b1">a</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <a href="toprealb.html#K10" title="TOPREALB:func.10">c[-10]</a> </span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c58">aX1</font></span>)</span> holds  <span class="p1">(<span class="default"><a href="toprealb.html#K15" title="TOPREALB:func.15">Circle2IntervalL</a>  <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="toprealb.html#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <a href="toprealb.html#K10" title="TOPREALB:func.10">c[-10]</a> </span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c58">aX1</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c48">Af</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c53">c</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c62">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c63">a</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <a href="toprealb.html#K10" title="TOPREALB:func.10">c[-10]</a> </span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c58">aX1</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c64">b</font> = <font color="Maroon" title="c63">a</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <a href="toprealb.html#K10" title="TOPREALB:func.10">c[-10]</a> </span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T2" title="TOPMETR:th.2">TOPMETR:2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c65">x</font>, <font color="Maroon" title="c66">y</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:109_2"/><i><font color="Green" title="E105">A14</font></i>: 
<font color="Maroon" title="c64">b</font> <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"><font color="Maroon" title="c65">x</font>,<font color="Maroon" title="c66">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>and </b><br/><a NAME="E4:109_2"/>
( <font color="Maroon" title="c66">y</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 implies <a href="toprealb.html#K15" title="TOPREALB:func.15">Circle2IntervalL</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c64">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <font color="Maroon" title="c65">x</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span> )
 <b>and </b><br/><a NAME="E5:109_2"/><i><font color="Green" title="E106">A15</font></i>: 
( <font color="Maroon" title="c66">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0 implies <a href="toprealb.html#K15" title="TOPREALB:func.15">Circle2IntervalL</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c64">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <font color="Maroon" title="c65">x</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="toprealb.html#D14" target="_self" title="TOPREALB:def.14">Def14</a></i>;<br/>
<a NAME="E6:109_2"/>
<font color="Maroon" title="c63">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c58">aX1</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E7:109"><i><font color="Green" title="E99">A3</font></i></a></i>;<br/>
<a NAME="E7:109_2"/><b>then </b><i><font color="Green" title="E107">A16</font></i>: 
 ex <font color="Olive" title="b1">q</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Maroon" title="c63">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> &amp; <font color="Olive" title="b1">q</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"><a href="toprealb.html#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <a href="toprealb.html#K10" title="TOPREALB:func.10">c[-10]</a> </span>)</span> &amp; 0 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 
<b>by </b><i><a class="txt" href="toprealb.html#E1:109"><i><font color="Green" title="E97">A1</font></i></a></i>;<br/>
<a NAME="E8:109_2"/><i><font color="Green" title="E108">A17</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c65">x</font>,<font color="Maroon" title="c66">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c65">x</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E9:109_2"/><i><font color="Green" title="E109">A18</font></i>: 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c65">x</font>,<font color="Maroon" title="c66">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 
<b>by </b><i><a class="txt" href="toprealb.html#E3:109_2"><i><font color="Green" title="E105">A14</font></i></a>, <a class="ref" href="toprealb.html#T29" target="_self" title="TOPREALB:th.29">Th29</a></i>;<br/>
<a NAME="E10:109_2"/>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c65">x</font>,<font color="Maroon" title="c66">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<b>by </b><i><a class="txt" href="toprealb.html#E3:109_2"><i><font color="Green" title="E105">A14</font></i></a>, <a class="ref" href="toprealb.html#T29" target="_self" title="TOPREALB:th.29">Th29</a></i>;<br/>
<a NAME="E11:109_2"/><b>then </b><i><font color="Green" title="E110">A19</font></i>: 
<font color="Maroon" title="c65">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c44">Q</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E8:109_2"><i><font color="Green" title="E108">A17</font></i></a>, <a class="txt" href="toprealb.html#E9:109_2"><i><font color="Green" title="E109">A18</font></i></a>, <a class="ref" href="rcomp_2.html#T4" title="RCOMP_2:th.4">RCOMP_2:4</a></i>;<br/>
<a NAME="E12:109_2"/><b>then </b>
<a href="sin_cos6.html#K4" title="SIN_COS6:func.4">arccos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c65">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c53">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c65">x</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<a NAME="E13:109_2"/><b>then </b><i><font color="Green" title="E111">A20</font></i>: 
<a href="sin_cos6.html#K4" title="SIN_COS6:func.4">arccos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c65">x</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="c53">c</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E11:109_2"><i><font color="Green" title="E110">A19</font></i></a>, <a class="txt" href="toprealb.html#E147"><i><font color="Green" title="E92">Lm56</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="toprealb.html#K15" title="TOPREALB:func.15">Circle2IntervalL</a>  <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="toprealb.html#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <a href="toprealb.html#K10" title="TOPREALB:func.10">c[-10]</a> </span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c58">aX1</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c63">a</font> = 
1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K6" title="SIN_COS6:func.6">arccos</a> <font color="Maroon" title="c65">x</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E3:109_2"><i><font color="Green" title="E105">A14</font></i></a>, <a class="txt" href="toprealb.html#E5:109_2"><i><font color="Green" title="E106">A15</font></i></a>, <a class="txt" href="toprealb.html#E7:109_2"><i><font color="Green" title="E107">A16</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos6.html#K4" title="SIN_COS6:func.4">arccos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c65">x</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="sin_cos6.html#D4" title="SIN_COS6:def.4">SIN_COS6:def 4</a></i>
<br/>.= 
1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="sin_cos6.html#K4" title="SIN_COS6:func.4">arccos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c65">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100" title="XCMPLX_1:th.100">XCMPLX_1:100</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="sin_cos6.html#K4" title="SIN_COS6:func.4">arccos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c65">x</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1

<br/>.= 
<span class="p1">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="sin_cos.html#K35" title="SIN_COS:func.35">PI</a> </span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="sin_cos6.html#K4" title="SIN_COS6:func.4">arccos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c65">x</font></span>)</span>
<b>by </b><i><a class="ref" href="jordan16.html#D3" title="JORDAN16:def.3">JORDAN16:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c48">Af</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="sin_cos6.html#K4" title="SIN_COS6:func.4">arccos</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c65">x</font></span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E13:109_2"><i><font color="Green" title="E111">A20</font></i></a>, <a class="txt" href="toprealb.html#E148"><i><font color="Green" title="E93">Lm57</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c48">Af</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c53">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c65">x</font></span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E11:109_2"><i><font color="Green" title="E110">A19</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c48">Af</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c53">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c65">x</font>,<font color="Maroon" title="c66">y</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>
<br/>.= 
<font color="Maroon" title="c48">Af</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c53">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c38">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c63">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E3:109_2"><i><font color="Green" title="E105">A14</font></i></a>, <a class="ref" href="pscomp_1.html#D28" title="PSCOMP_1:def.28">PSCOMP_1:def 28</a></i>
<br/>.= 
<font color="Maroon" title="c48">Af</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c53">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c62">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c63">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c48">Af</font> <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="c53">c</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c62">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c63">a</font></span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E13:109"><i><font color="Green" title="E101">A5</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c48">Af</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c53">c</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c62">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c63">a</font>
<b>by </b><i><a class="txt" href="toprealb.html#E13:109"><i><font color="Green" title="E101">A5</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E22:109"/>
<a href="toprealb.html#K15" title="TOPREALB:func.15">Circle2IntervalL</a>  <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K11" title="TOPREALB:func.11">Topen_unit_circle</a> <a href="toprealb.html#K10" title="TOPREALB:func.10">c[-10]</a> </span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c56">aX1</font></span>)</span> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="toprealb.html#E13:109"><i><font color="Green" title="E101">A5</font></i></a>, <a class="txt" href="toprealb.html#E20:109"><i><font color="Green" title="E104">A13</font></i></a>, <a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>


</div>
