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

<b>set </b><font color="Maroon" title="c19">f</font> = <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_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c18">r</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span>;<br/>
<b>set </b><font color="Maroon" title="c20">g</font> = <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_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default">0,1</span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span>;<br/>
<a NAME="E1:86_1_1"/><i><font color="Green" title="E55">A1</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> <span class="p2"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c18">r</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c18">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c18">r</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span>
 
<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/>
<a NAME="E2:86_1_1"/><i><font color="Green" title="E56">A2</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> <span class="p2"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default">0,1</span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default">0,1</span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span>
 
<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>let </b><font color="Maroon" title="c21">x</font>, <font color="Maroon" title="c22">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="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E3:86_1_1"/><i><font color="Green" title="E57">A3</font></i>: 
<font color="Maroon" title="c21">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"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c18">r</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c18">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:86_1_1"/><i><font color="Green" title="E58">A4</font></i>: 
<font color="Maroon" title="c22">y</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"><a href="toprealb.html#K12" title="TOPREALB:func.12">CircleMap</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c18">r</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c18">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span></span>)</span>
 <b>and </b><br/><a NAME="E5:86_1_1"/><i><font color="Green" title="E59">A5</font></i>: 
<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> <span class="p2"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c18">r</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c18">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <span class="p2"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c18">r</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c18">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">y</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c23">x</font> = <font color="Maroon" title="c21">x</font>, <font color="Maroon" title="c24">y</font> = <font color="Maroon" title="c22">y</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#E1:86_1_1"><i><font color="Green" title="E55">A1</font></i></a>, <a class="txt" href="toprealb.html#E3:86_1_1"><i><font color="Green" title="E57">A3</font></i></a>, <a class="txt" href="toprealb.html#E4:86_1_1"><i><font color="Green" title="E58">A4</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c25">x1</font> =  <a href="int_1.html#K4" title="INT_1:func.4">frac</a> <font color="Maroon" title="c23">x</font>;<br/>
<b>set </b><font color="Maroon" title="c26">y1</font> =  <a href="int_1.html#K4" title="INT_1:func.4">frac</a> <font color="Maroon" title="c24">y</font>;<br/>
<a NAME="E7:86_1_1"/><i><font color="Green" title="E60">A6</font></i>: 
 <a href="int_1.html#K4" title="INT_1:func.4">frac</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">x</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="c23">x</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#D6" title="INT_1:def.6">INT_1:def 6</a></i>;<br/>
<a NAME="E8:86_1_1"/><i><font color="Green" title="E61">A7</font></i>: 
 <a href="int_1.html#K4" title="INT_1:func.4">frac</a> <font color="Maroon" title="c24">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">y</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="c24">y</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#D6" title="INT_1:def.6">INT_1:def 6</a></i>;<br/>
<a NAME="E9:86_1_1"/>
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="int_1.html#K4" title="INT_1:func.4">frac</a> <font color="Maroon" title="c23">x</font> &amp;  <a href="int_1.html#K4" title="INT_1:func.4">frac</a> <font color="Maroon" title="c23">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 
<b>by </b><i><a class="ref" href="int_1.html#T69" title="INT_1:th.69">INT_1:69</a></i>;<br/>
<a NAME="E10:86_1_1"/><b>then </b><i><font color="Green" title="E62">A8</font></i>: 
 <a href="int_1.html#K4" title="INT_1:func.4">frac</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default">0,1</span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_2.html#T3" title="RCOMP_2:th.3">RCOMP_2:3</a></i>;<br/>
<a NAME="E11:86_1_1"/>
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="int_1.html#K4" title="INT_1:func.4">frac</a> <font color="Maroon" title="c24">y</font> &amp;  <a href="int_1.html#K4" title="INT_1:func.4">frac</a> <font color="Maroon" title="c24">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 
<b>by </b><i><a class="ref" href="int_1.html#T69" title="INT_1:th.69">INT_1:69</a></i>;<br/>
<a NAME="E12:86_1_1"/><b>then </b><i><font color="Green" title="E63">A9</font></i>: 
 <a href="int_1.html#K4" title="INT_1:func.4">frac</a> <font color="Maroon" title="c24">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default">0,1</span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_2.html#T3" title="RCOMP_2:th.3">RCOMP_2:3</a></i>;<br/>
<a NAME="E13:86_1_1"/><i><font color="Green" title="E64">A10</font></i>: 
( <font color="Maroon" title="c18">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">x</font> &amp; <font color="Maroon" title="c23">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 )
 
<b>by </b><i><a class="txt" href="toprealb.html#E1:86_1_1"><i><font color="Green" title="E55">A1</font></i></a>, <a class="txt" href="toprealb.html#E3:86_1_1"><i><font color="Green" title="E57">A3</font></i></a>, <a class="ref" href="rcomp_2.html#T3" title="RCOMP_2:th.3">RCOMP_2:3</a></i>;<br/>
<a NAME="E14:86_1_1"/><i><font color="Green" title="E65">A11</font></i>: 
( <font color="Maroon" title="c18">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">y</font> &amp; <font color="Maroon" title="c24">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 )
 
<b>by </b><i><a class="txt" href="toprealb.html#E1:86_1_1"><i><font color="Green" title="E55">A1</font></i></a>, <a class="txt" href="toprealb.html#E4:86_1_1"><i><font color="Green" title="E58">A4</font></i></a>, <a class="ref" href="rcomp_2.html#T3" title="RCOMP_2:th.3">RCOMP_2:3</a></i>;<br/>
<i><font color="Green" title="E66">A12</font></i>: <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> <span class="p2"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c18">r</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c18">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> = 
<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="c23">x</font>
<b>by </b><i><a class="txt" href="toprealb.html#E3:86_1_1"><i><font color="Green" title="E57">A3</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</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> <span class="p1">(<span class="default"><font color="Maroon" title="c23">x</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="c23">x</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/>.= 
<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> <span class="p2"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default">0,1</span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="int_1.html#K4" title="INT_1:func.4">frac</a> <font color="Maroon" title="c23">x</font></span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E2:86_1_1"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="toprealb.html#E7:86_1_1"><i><font color="Green" title="E60">A6</font></i></a>, <a class="txt" href="toprealb.html#E10:86_1_1"><i><font color="Green" title="E62">A8</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
;<br/>
<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> <span class="p2"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c18">r</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c18">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">y</font> = 
<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="c24">y</font>
<b>by </b><i><a class="txt" href="toprealb.html#E4:86_1_1"><i><font color="Green" title="E58">A4</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</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> <span class="p1">(<span class="default"><font color="Maroon" title="c24">y</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="c24">y</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/>.= 
<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> <span class="p2"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default">0,1</span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="int_1.html#K4" title="INT_1:func.4">frac</a> <font color="Maroon" title="c24">y</font></span>)</span>
<b>by </b><i><a class="txt" href="toprealb.html#E2:86_1_1"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="toprealb.html#E8:86_1_1"><i><font color="Green" title="E61">A7</font></i></a>, <a class="txt" href="toprealb.html#E12:86_1_1"><i><font color="Green" title="E63">A9</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
;<br/>
<a NAME="E17:86_1_1"/><b>then </b>
 <a href="int_1.html#K4" title="INT_1:func.4">frac</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_1.html#K4" title="INT_1:func.4">frac</a> <font color="Maroon" title="c24">y</font>
 
<b>by </b><i><a class="txt" href="toprealb.html#E2:86_1_1"><i><font color="Green" title="E56">A2</font></i></a>, <a class="txt" href="toprealb.html#E5:86_1_1"><i><font color="Green" title="E59">A5</font></i></a>, <a class="txt" href="toprealb.html#E10:86_1_1"><i><font color="Green" title="E62">A8</font></i></a>, <a class="txt" href="toprealb.html#E12:86_1_1"><i><font color="Green" title="E63">A9</font></i></a>, <a class="txt" href="toprealb.html#E15:86_1_1"><i><font color="Green" title="E66">A12</font></i></a>, <a class="txt" href="toprealb.html#E77"><i><font color="Green" title="E54">Lm25</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E18:86_1_1"/>
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">y</font>
 <b>by </b><i><a class="txt" href="toprealb.html#E13:86_1_1"><i><font color="Green" title="E64">A10</font></i></a>, <a class="txt" href="toprealb.html#E14:86_1_1"><i><font color="Green" title="E65">A11</font></i></a>, <a class="ref" href="topreala.html#T7" title="TOPREALA:th.7">TOPREALA:7</a></i>;<br/>


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