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

<b>let </b><font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>set </b><font color="Maroon">c<sub>20</sub></font> =  <a href="jordan16.html#K1">AffineMap</a> 1,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>21</sub></font> = <span class="p1">(<span class="default"><a href="jordan16.html#K1">AffineMap</a> 1,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>;<br/>
<a NAME="E1:91"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jordan16.html#K1">AffineMap</a> 1,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:91"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan16.html#K1">AffineMap</a> 1,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <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="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<b>thus </b><a NAME="E3:91"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan16.html#K1">AffineMap</a> 1,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <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>19</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:91_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan16.html#K1">AffineMap</a> 1,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <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> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>22</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="E1:91_1_1"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan16.html#K1">AffineMap</a> 1,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>23</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:91_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan16.html#K1">AffineMap</a> 1,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <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="E4:91_1_1"/><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan16.html#K1">AffineMap</a> 1,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>23</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="toprealb.html#E3:91_1_1"><i><font color="Green">E59</font></i></a></i>;<br/>
<i><font color="Green">E61</font></i>: <font color="Maroon">c<sub>22</sub></font> = 
<span class="p1">(<span class="default"><a href="jordan16.html#K1">AffineMap</a> 1,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="txt" href="toprealb.html#E3:91_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="toprealb.html#E4:91_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="jordan16.html#D3">JORDAN16:def 3</a></i>
;<br/>
<a NAME="E7:91_1_1"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 )
 
<b>by </b><i><a class="txt" href="toprealb.html#E2:91"><i><font color="Green">E58</font></i></a>, <a class="txt" href="toprealb.html#E3:91_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<a NAME="E8:91_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>24</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>18</sub></font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<b>hence </b><a NAME="E9:91_1_1"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>19</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#E6:91_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>22</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="E2:91_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>22</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E4:91_1"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 )
 
<b>by </b><i><a class="txt" href="toprealb.html#E2:91_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<a NAME="E5:91_1"/><b>then </b>
( <font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>23</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E6:91_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <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="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan16.html#K1">AffineMap</a> 1,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="jordan16.html#K1">AffineMap</a> 1,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="jordan16.html#D3">JORDAN16:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>23</sub></font>

;<br/>
<b>hence </b><a NAME="E8:91_1"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan16.html#K1">AffineMap</a> 1,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="toprealb.html#E2:91"><i><font color="Green">E58</font></i></a>, <a class="txt" href="toprealb.html#E6:91_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>


</div>
