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

<a NAME="E1:80_1_1"/><i><font color="Green" title="E51">A1</font></i>: 
 <a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p1">(<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>,0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="toprealb.html#K6" title="TOPREALB:func.6">R^1</a> <span class="p1">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</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>,0</span>)</span>
 
;<br/>
<a NAME="E2:80_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="sin_cos.html#D20" title="SIN_COS:def.20">SIN_COS:def 20</a></i>;<br/>
<a NAME="E3:80_1_1"/><b>then </b><i><font color="Green" title="E52">A2</font></i>: 
<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="p1">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a> 
 
<b>by </b><i><a class="ref" href="toprealb.html#T6" target="_self" title="TOPREALB:th.6">Th6</a></i>;<br/>
<a NAME="E4:80_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="sin_cos.html#D22" title="SIN_COS:def.22">SIN_COS:def 22</a></i>;<br/>
<a NAME="E5:80_1_1"/><b>then </b><i><font color="Green" title="E53">A3</font></i>: 
<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="p1">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a> 
 
<b>by </b><i><a class="ref" href="toprealb.html#T6" target="_self" title="TOPREALB:th.6">Th6</a></i>;<br/>
<b>set </b><font color="Maroon" title="c18">sR</font> =  <a href="toprealb.html#K6" title="TOPREALB:func.6">R^1</a> <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> ;<br/>
<b>set </b><font color="Maroon" title="c19">cR</font> =  <a href="toprealb.html#K6" title="TOPREALB:func.6">R^1</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c20">sR</font> =  <a href="toprealb.html#K6" title="TOPREALB:func.6">R^1</a> <a href="sin_cos.html#K19" title="SIN_COS:func.19">sin</a> , <font color="Maroon" title="c21">cR</font> =  <a href="toprealb.html#K6" title="TOPREALB:func.6">R^1</a> <a href="sin_cos.html#K22" title="SIN_COS:func.22">cos</a>  as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a> ,<a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <b>by </b><i><a class="txt" href="toprealb.html#E25"><i><font color="Green" title="E17">Lm11</font></i></a>, <a class="txt" href="toprealb.html#E26"><i><font color="Green" title="E18">Lm12</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c22">l</font> =  <a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p1">(<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>,0 as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a> ,<a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a>  <b>by </b><i><a class="ref" href="toprealb.html#T8" target="_self" title="TOPREALB:th.8">Th8</a></i>;<br/>
<a NAME="E8:80_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</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>,0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:80_1_1"/><b>then </b><i><font color="Green" title="E54">A4</font></i>: 
<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="p1">(<span class="default"><a href="toprealb.html#K5" title="TOPREALB:func.5">R^1</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</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>,0</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a> 
 
<b>by </b><i><a class="ref" href="toprealb.html#T6" target="_self" title="TOPREALB:th.6">Th6</a></i>;<br/>
<a NAME="E10:80_1_1"/><i><font color="Green" title="E55">A5</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</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>,0</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="jordan16.html#T32" title="JORDAN16:th.32">JORDAN16:32</a></i>;<br/>
<b>set </b><font color="Maroon" title="c23">s</font> = <font color="Maroon" title="c20">sR</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">l</font>;<br/>
<b>set </b><font color="Maroon" title="c24">c</font> = <font color="Maroon" title="c21">cR</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">l</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c25">g</font> =  <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topalg_2.html#K3" title="TOPALG_2:func.3">R^1</a> ,<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="topreala.html#T28" title="TOPREALA:th.28">TOPREALA:28</a></i>;<br/>
<a NAME="E12:80_1_1"/>
for <font color="Olive" title="b1">p</font> being  <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> <br/> for <font color="Olive" title="b2">V</font> being  <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>  st <font color="Maroon" title="c25">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">V</font> &amp; <font color="Olive" title="b2">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/> ex <font color="Olive" title="b3">W</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 <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">W</font> &amp; <font color="Olive" title="b3">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c25">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b3">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c26">p</font> be   <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> ;<br/><b>let </b><font color="Maroon" title="c27">V</font> be   <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>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:80_1_1_1"/><i><font color="Green" title="E56">A6</font></i>: 
<font color="Maroon" title="c25">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">V</font>
 <b>and </b><br/><a NAME="E2:80_1_1_1"/><i><font color="Green" title="E57">A7</font></i>: 
<font color="Maroon" title="c27">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<a NAME="E3:80_1_1_1"/><i><font color="Green" title="E58">A8</font></i>: 
<font color="Maroon" title="c27">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c27">V</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E2:80_1_1_1"><i><font color="Green" title="E57">A7</font></i></a>, <a class="ref" href="tops_1.html#T55" title="TOPS_1:th.55">TOPS_1:55</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c28">e</font> = <font color="Maroon" title="c25">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">p</font> 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#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13" title="TOPREAL3:th.13">TOPREAL3:13</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c29">r</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="E6:80_1_1_1"/><i><font color="Green" title="E59">A9</font></i>: 
<font color="Maroon" title="c29">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>and </b><br/><a NAME="E7:80_1_1_1"/><i><font color="Green" title="E60">A10</font></i>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c28">e</font>,<font color="Maroon" title="c29">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c27">V</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E1:80_1_1_1"><i><font color="Green" title="E56">A6</font></i></a>, <a class="txt" href="toprealb.html#E3:80_1_1_1"><i><font color="Green" title="E58">A8</font></i></a>, <a class="ref" href="goboard6.html#T8" title="GOBOARD6:th.8">GOBOARD6:8</a></i>;<br/>
<b>set </b><font color="Maroon" title="c30">A</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"><span class="p4">(<span class="default"><font color="Maroon" title="c25">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">p</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c25">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">p</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>;<br/>
<b>set </b><font color="Maroon" title="c31">B</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"><span class="p4">(<span class="default"><font color="Maroon" title="c25">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">p</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c25">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">p</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>;<br/>
<b>set </b><font color="Maroon" title="c32">F</font> = 1,2 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</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"><span class="p4">(<span class="default"><font color="Maroon" title="c25">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">p</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c25">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">p</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>,<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"><span class="p4">(<span class="default"><font color="Maroon" title="c25">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">p</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c25">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">p</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>;<br/>
<a NAME="E8:80_1_1_1"/>
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="square_1.html#T84" title="SQUARE_1:th.84">SQUARE_1:84</a></i>;<br/>
<a NAME="E9:80_1_1_1"/><b>then </b><i><font color="Green" title="E61">A11</font></i>: 
<font color="Maroon" title="c29">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="toprealb.html#E6:80_1_1_1"><i><font color="Green" title="E59">A9</font></i></a>, <a class="ref" href="real_2.html#T127" title="REAL_2:th.127">REAL_2:127</a></i>;<br/>
<a NAME="E10:80_1_1_1"/><i><font color="Green" title="E62">A12</font></i>: 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default">1,2 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c25">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">p</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c29">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c25">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">p</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c29">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>,<span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c25">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">p</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c29">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c25">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">p</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c29">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c28">e</font>,<font color="Maroon" title="c29">r</font>
 
<b>by </b><i><a class="ref" href="topreal6.html#T49" title="TOPREAL6:th.49">TOPREAL6:49</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c33">A</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"><span class="p4">(<span class="default"><font color="Maroon" title="c25">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">p</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c25">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">p</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>, <font color="Maroon" title="c34">B</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"><span class="p4">(<span class="default"><font color="Maroon" title="c25">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">p</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c25">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">p</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 2</span>)</span></span>)</span></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 <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/>
<a NAME="E12:80_1_1_1"/><i><font color="Green" title="E63">A13</font></i>: 
<font color="Maroon" title="c34">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="jordan6.html#T46" title="JORDAN6:th.46">JORDAN6:46</a></i>;<br/>
<a NAME="E13:80_1_1_1"/><i><font color="Green" title="E64">A14</font></i>: 
<font color="Maroon" title="c25">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">p</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">cR</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">p</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">sR</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">p</font></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 
<b>by </b><i><a class="txt" href="toprealb.html#E72"><i><font color="Green" title="E50">Lm22</font></i></a></i>;<br/>
<a NAME="E14:80_1_1_1"/><b>then </b><i><font color="Green" title="E65">A15</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c25">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">p</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">sR</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">p</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E15:80_1_1_1"/>
<font color="Maroon" title="c20">sR</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#E3:80_1_1"><i><font color="Green" title="E52">A2</font></i></a>, <a class="ref" href="topmetr.html#T7" title="TOPMETR:th.7">TOPMETR:7</a></i>;<br/>
<a NAME="E16:80_1_1_1"/><b>then </b><i><font color="Green" title="E66">A16</font></i>: 
<font color="Maroon" title="c20">sR</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">l</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#E1:80_1_1"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="toprealb.html#E9:80_1_1"><i><font color="Green" title="E54">A4</font></i></a>, <a class="txt" href="toprealb.html#E10:80_1_1"><i><font color="Green" title="E55">A5</font></i></a>, <a class="ref" href="toprealb.html#T5" target="_self" title="TOPREALB:th.5">Th5</a>, <a class="ref" href="tops_2.html#T58" title="TOPS_2:th.58">TOPS_2:58</a></i>;<br/>
<a NAME="E17:80_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">sR</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">B</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E9:80_1_1_1"><i><font color="Green" title="E61">A11</font></i></a>, <a class="txt" href="toprealb.html#E14:80_1_1_1"><i><font color="Green" title="E65">A15</font></i></a>, <a class="ref" href="topreal6.html#T20" title="TOPREAL6:th.20">TOPREAL6:20</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c35">Ws</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> <b> such that </b><br/><a NAME="E19:80_1_1_1"/><i><font color="Green" title="E67">A17</font></i>: 
<font color="Maroon" title="c26">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c35">Ws</font>
 <b>and </b><br/><a NAME="E20:80_1_1_1"/><i><font color="Green" title="E68">A18</font></i>: 
<font color="Maroon" title="c35">Ws</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E21:80_1_1_1"/><i><font color="Green" title="E69">A19</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">sR</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c35">Ws</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c34">B</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E12:80_1_1_1"><i><font color="Green" title="E63">A13</font></i></a>, <a class="txt" href="toprealb.html#E16:80_1_1_1"><i><font color="Green" title="E66">A16</font></i></a>, <a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></i>;<br/>
<a NAME="E22:80_1_1_1"/><i><font color="Green" title="E70">A20</font></i>: 
<font color="Maroon" title="c33">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="jordan6.html#T46" title="JORDAN6:th.46">JORDAN6:46</a></i>;<br/>
<a NAME="E23:80_1_1_1"/><i><font color="Green" title="E71">A21</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c25">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">p</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">cR</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">p</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E13:80_1_1_1"><i><font color="Green" title="E64">A14</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E24:80_1_1_1"/>
<font color="Maroon" title="c21">cR</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#E5:80_1_1"><i><font color="Green" title="E53">A3</font></i></a>, <a class="ref" href="topmetr.html#T7" title="TOPMETR:th.7">TOPMETR:7</a></i>;<br/>
<a NAME="E25:80_1_1_1"/><b>then </b><i><font color="Green" title="E72">A22</font></i>: 
<font color="Maroon" title="c21">cR</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">l</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#E1:80_1_1"><i><font color="Green" title="E51">A1</font></i></a>, <a class="txt" href="toprealb.html#E9:80_1_1"><i><font color="Green" title="E54">A4</font></i></a>, <a class="txt" href="toprealb.html#E10:80_1_1"><i><font color="Green" title="E55">A5</font></i></a>, <a class="ref" href="toprealb.html#T5" target="_self" title="TOPREALB:th.5">Th5</a>, <a class="ref" href="tops_2.html#T58" title="TOPS_2:th.58">TOPS_2:58</a></i>;<br/>
<a NAME="E26:80_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c21">cR</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c33">A</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E9:80_1_1_1"><i><font color="Green" title="E61">A11</font></i></a>, <a class="txt" href="toprealb.html#E23:80_1_1_1"><i><font color="Green" title="E71">A21</font></i></a>, <a class="ref" href="topreal6.html#T20" title="TOPREAL6:th.20">TOPREAL6:20</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c36">Wc</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> <b> such that </b><br/><a NAME="E28:80_1_1_1"/><i><font color="Green" title="E73">A23</font></i>: 
<font color="Maroon" title="c26">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c36">Wc</font>
 <b>and </b><br/><a NAME="E29:80_1_1_1"/><i><font color="Green" title="E74">A24</font></i>: 
<font color="Maroon" title="c36">Wc</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E30:80_1_1_1"/><i><font color="Green" title="E75">A25</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">cR</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c36">Wc</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c33">A</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E22:80_1_1_1"><i><font color="Green" title="E70">A20</font></i></a>, <a class="txt" href="toprealb.html#E25:80_1_1_1"><i><font color="Green" title="E72">A22</font></i></a>, <a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></i>;<br/>
<b>set </b><font color="Maroon" title="c37">W</font> = <font color="Maroon" title="c35">Ws</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c36">Wc</font>;<br/>
<b>take </b>
<font color="Maroon" title="c35">Ws</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c36">Wc</font>
;<br/>

<b>thus </b><a NAME="E31:80_1_1_1"/>
<font color="Maroon" title="c26">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c35">Ws</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c36">Wc</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E19:80_1_1_1"><i><font color="Green" title="E67">A17</font></i></a>, <a class="txt" href="toprealb.html#E28:80_1_1_1"><i><font color="Green" title="E73">A23</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>

<b>thus </b><a NAME="E32:80_1_1_1"/>
<font color="Maroon" title="c35">Ws</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c36">Wc</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#E20:80_1_1_1"><i><font color="Green" title="E68">A18</font></i></a>, <a class="txt" href="toprealb.html#E29:80_1_1_1"><i><font color="Green" title="E74">A24</font></i></a>, <a class="ref" href="tops_1.html#T38" title="TOPS_1:th.38">TOPS_1:38</a></i>;<br/>

<b>let </b><font color="Maroon" title="c38">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="E33:80_1_1_1"/>
<font color="Maroon" title="c38">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c35">Ws</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c36">Wc</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c39">x</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="E35:80_1_1_1"/><i><font color="Green" title="E76">A26</font></i>: 
<font color="Maroon" title="c39">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c35">Ws</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c36">Wc</font>
 <b>and </b><br/><a NAME="E36:80_1_1_1"/><i><font color="Green" title="E77">A27</font></i>: 
<font color="Maroon" title="c38">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c39">x</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<a NAME="E37:80_1_1_1"/><i><font color="Green" title="E78">A28</font></i>: 
<font color="Maroon" title="c25">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c39">x</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">cR</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c39">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">sR</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c39">x</font></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 
<b>by </b><i><a class="txt" href="toprealb.html#E72"><i><font color="Green" title="E50">Lm22</font></i></a></i>;<br/>
<a NAME="E38:80_1_1_1"/>
<font color="Maroon" title="c39">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c35">Ws</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E35:80_1_1_1"><i><font color="Green" title="E76">A26</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E39:80_1_1_1"/><b>then </b><i><font color="Green" title="E79">A29</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">sR</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c39">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">sR</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c35">Ws</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>
<a NAME="E40:80_1_1_1"/>
<font color="Maroon" title="c39">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c36">Wc</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E35:80_1_1_1"><i><font color="Green" title="E76">A26</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E41:80_1_1_1"/><b>then </b><i><font color="Green" title="E80">A30</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">cR</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c39">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">cR</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c36">Wc</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>
<a NAME="E42:80_1_1_1"/>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">cR</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c39">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">sR</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c39">x</font></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1,2 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">cR</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c39">x</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">sR</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c39">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topreala.html#T49" title="TOPREALA:th.49">TOPREALA:49</a></i>;<br/>
<a NAME="E43:80_1_1_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">cR</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c39">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">sR</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c39">x</font></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default">1,2 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c25">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">p</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c29">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c25">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">p</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c29">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>,<span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c25">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">p</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c29">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c25">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">p</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c29">r</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> 2</span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="toprealb.html#E21:80_1_1_1"><i><font color="Green" title="E69">A19</font></i></a>, <a class="txt" href="toprealb.html#E30:80_1_1_1"><i><font color="Green" title="E75">A25</font></i></a>, <a class="txt" href="toprealb.html#E39:80_1_1_1"><i><font color="Green" title="E79">A29</font></i></a>, <a class="txt" href="toprealb.html#E41:80_1_1_1"><i><font color="Green" title="E80">A30</font></i></a>, <a class="ref" href="hilbert3.html#T11" title="HILBERT3:th.11">HILBERT3:11</a></i>;<br/>
<a NAME="E44:80_1_1_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">cR</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c39">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">sR</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c22">l</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c39">x</font></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c28">e</font>,<font color="Maroon" title="c29">r</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E10:80_1_1_1"><i><font color="Green" title="E62">A12</font></i></a></i>;<br/>
<b>hence </b><a NAME="E45:80_1_1_1"/>
<font color="Maroon" title="c38">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">V</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E7:80_1_1_1"><i><font color="Green" title="E60">A10</font></i></a>, <a class="txt" href="toprealb.html#E36:80_1_1_1"><i><font color="Green" title="E77">A27</font></i></a>, <a class="txt" href="toprealb.html#E37:80_1_1_1"><i><font color="Green" title="E78">A28</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:80_1_1"/><b>then </b>
<font color="Maroon" title="c25">g</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="jgraph_2.html#T20" title="JGRAPH_2:th.20">JGRAPH_2:20</a></i>;<br/>
<b>hence </b><a NAME="E14:80_1_1"/>
 <a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="topmetr.html#T8" title="TOPMETR:th.8">TOPMETR:8</a></i>;<br/>


</div>
