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

<b>set </b><font color="Maroon">f</font> = <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_2.html#K1">.[</a></span>;<br/>
<b>set </b><font color="Maroon">g</font> = <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_2.html#K1">.[</a></span>;<br/>
<a NAME="E1:86_1_1"/><i><font color="Green">E61</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_2.html#K1">.[</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_2.html#K1">.[</a></span>
 
<b>by </b><i><a class="txt" href="toprealb.html#E6"><i><font color="Green">Lemma43</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E2:86_1_1"/><i><font color="Green">E66</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_2.html#K1">.[</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_2.html#K1">.[</a></span>
 
<b>by </b><i><a class="txt" href="toprealb.html#E6"><i><font color="Green">Lemma43</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E3:86_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_2.html#K1">.[</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:86_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_2.html#K1">.[</a></span></span>)</span>
 <b>and </b><br/><a NAME="E5:86_1_1"/><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_2.html#K1">.[</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_2.html#K1">.[</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font>, <font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, , </i>;<br/>
<b>set </b><font color="Maroon">x1</font> =  <a href="int_1.html#K4">frac</a> <font color="Maroon">x</font>;<br/>
<b>set </b><font color="Maroon">y1</font> =  <a href="int_1.html#K4">frac</a> <font color="Maroon">y</font>;<br/>
<a NAME="E7:86_1_1"/><i><font color="Green">E72</font></i>: 
 <a href="int_1.html#K4">frac</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x</font></span><a href="int_1.html#K1">/]</a></span>
 
<b>by </b><i><a class="ref" href="int_1.html#D6">INT_1:def 6</a></i>;<br/>
<a NAME="E8:86_1_1"/><i><font color="Green">E73</font></i>: 
 <a href="int_1.html#K4">frac</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">y</font></span><a href="int_1.html#K1">/]</a></span>
 
<b>by </b><i><a class="ref" href="int_1.html#D6">INT_1:def 6</a></i>;<br/>
<a NAME="E9:86_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="int_1.html#K4">frac</a> <font color="Maroon">x</font> &amp;  <a href="int_1.html#K4">frac</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 
<b>by </b><i><a class="ref" href="int_1.html#T69">INT_1:69</a></i>;<br/>
<a NAME="E10:86_1_1"/><b>then </b><i><font color="Green">E74</font></i>: 
 <a href="int_1.html#K4">frac</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_2.html#K1">.[</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/>
<a NAME="E11:86_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="int_1.html#K4">frac</a> <font color="Maroon">y</font> &amp;  <a href="int_1.html#K4">frac</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 
<b>by </b><i><a class="ref" href="int_1.html#T69">INT_1:69</a></i>;<br/>
<a NAME="E12:86_1_1"/><b>then </b><i><font color="Green">E75</font></i>: 
 <a href="int_1.html#K4">frac</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_2.html#K1">.[</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/>
<a NAME="E13:86_1_1"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> 1 )
 
<b>by </b><i>, , <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/>
<a NAME="E14:86_1_1"/><i><font color="Green">E115</font></i>: 
( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> 1 )
 
<b>by </b><i>, , <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/>
<i><font color="Green">E116</font></i>: <span class="p1">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_2.html#K1">.[</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">x</font></span><a href="int_1.html#K1">/]</a></span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_2.html#K1">.[</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="int_1.html#K4">frac</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="toprealb.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
;<br/>
<span class="p1">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">r</font>,<span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_2.html#K1">.[</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> = 
<a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">y</font></span><a href="int_1.html#K1">/]</a></span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_2.html#K1">.[</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="int_1.html#K4">frac</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, <a class="txt" href="toprealb.html#E1:86_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="toprealb.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
;<br/>
<a NAME="E17:86_1_1"/><b>then </b>
 <a href="int_1.html#K4">frac</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="int_1.html#K4">frac</a> <font color="Maroon">y</font>
 
<b>by </b><i>, <a class="txt" href="toprealb.html#E12"><i><font color="Green">Lemma59</font></i></a>, <a class="ref" href="toprealb.html#T1" target="_self">Th1</a>, <a class="ref" href="toprealb.html#T2" target="_self">Th2</a>, <a class="txt" href="toprealb.html#E2:86_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="toprealb.html#E11"><i><font color="Green">Lemma54</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E18:86_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="topreala.html#T7">TOPREALA:7</a></i>;<br/>


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