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

<b>set </b><font color="Maroon">c<sub>19</sub></font> =  <a href="toprealb.html#K11" target="_self">Topen_unit_circle</a> <span class="p1">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>21</sub></font> = <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></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">c<sub>22</sub></font> = <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>23</sub></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">c<sub>18</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_2.html#K1">.[</a></span>;<br/>
<a NAME="E1:90_1_1"/><i><font color="Green">E57</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></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="ref" href="topreala.html#T20">TOPREALA:20</a></i>;<br/>
<a NAME="E2:90_1_1"/><i><font color="Green">E58</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_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 
<b>by </b><i><a class="txt" href="toprealb.html#E62"><i><font color="Green">Lemma43</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E3:90_1_1"/><i><font color="Green">E59</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K11" target="_self">Topen_unit_circle</a> <span class="p2">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" target="_self">Tunit_circle</a> 2</span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="toprealb.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E4:90_1_1"/><i><font color="Green">E60</font></i>: 
 <a href="relset_1.html#K5">rng</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_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K11" target="_self">Topen_unit_circle</a> <span class="p2">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>24</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:90_1_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</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_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:90_1_1_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>25</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:90_1_1_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>25</sub></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_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:90_1_1_1_1"/><i><font color="Green">E64</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_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="toprealb.html#E1:90_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:90_1_1_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>18</sub></font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E6:90_1_1_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></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="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/><a NAME="E7:90_1_1_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"><font color="Maroon">c<sub>18</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></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">c<sub>18</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></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#E62"><i><font color="Green">Lemma43</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</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">c<sub>18</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></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">c<sub>25</sub></font> = 
<a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="txt" href="toprealb.html#E1:90_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="toprealb.html#E2:90_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="toprealb.html#E3:90_1_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="toprealb.html#E2:90_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="toprealb.html#E1:90_1_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="toprealb.html#E3:90_1_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="toprealb.html#E4:90_1_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</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">c<sub>18</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></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">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="toprealb.html#E6:90_1_1_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><a NAME="E9:90_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="toprealb.html#E1:90_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="toprealb.html#E2:90_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="toprealb.html#E3:90_1_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="toprealb.html#E6:90_1_1_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="toprealb.html#E7:90_1_1_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E10:90_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="toprealb.html#E2:90_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="toprealb.html#E3:90_1_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:90_1_1_1"/><b>then </b>
not <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:90_1_1_1"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K11" target="_self">Topen_unit_circle</a> <span class="p2">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="toprealb.html#E3:90_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="toprealb.html#E1:90_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" target="_self">R^1</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span></span>)</span> = 
 <a href="toprealb.html#K5" target="_self">R^1</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>

;<br/>
<b>hence </b><a NAME="E6:90_1_1"/>
<a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> is   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="topalg_2.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="toprealb.html#K5" target="_self">R^1</a> <span class="p3"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="toprealb.html#K11" target="_self">Topen_unit_circle</a> <span class="p2">(<span class="default"><a href="toprealb.html#K12" target="_self">CircleMap</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="toprealb.html#E2:90_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="toprealb.html#E4:90_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>


</div>
