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

<b>let </b><font color="Maroon">c1</font> be  <a href="xcmplx_0.html#V1" target="_self">complex</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">c2</font> be  <a href="xcmplx_0.html#V1" target="_self">complex</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>thus </b><a NAME="E1:9_2_2"/>
( <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">z</font> <a href="xcmplx_0.html#K3" target="_self">*</a> <font color="Maroon">c1</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">z</font> <a href="xcmplx_0.html#K3" target="_self">*</a> <font color="Maroon">c2</font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">c1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_2_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:9_2_2_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E2:9_2_2_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">z</font> <a href="xcmplx_0.html#K3" target="_self">*</a> <font color="Maroon">c1</font> <a href="hidden.html#R1">=</a> 1
 <b>and </b><br/><a NAME="E3:9_2_2_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">z</font> <a href="xcmplx_0.html#K3" target="_self">*</a> <font color="Maroon">c2</font> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<a NAME="E4:9_2_2_1"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">z'</font> being <a href="xcmplx_0.html#V1" target="_self">complex</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">z</font> <a href="xcmplx_0.html#K3" target="_self">*</a> <font color="Olive">z'</font> <a href="hidden.html#R1">=</a> 1 holds <br/><font color="Olive">z'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p5">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p5">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z'</font> be  <a href="xcmplx_0.html#V1" target="_self">complex</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:9_2_2_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">z</font> <a href="xcmplx_0.html#K3" target="_self">*</a> <font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<b>consider </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">x2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">x'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">y'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:9_2_2_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="arytm_0.html#K5">*]</a></span>
 <b>and </b><br/><a NAME="E4:9_2_2_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">x'</font>,<font color="Maroon">y'</font></span><a href="arytm_0.html#K5">*]</a></span>
 <b>and </b><br/><a NAME="E5:9_2_2_1_1"/><i><font color="Green">E66</font></i>: 
1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x1</font>,<font color="Maroon">x'</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x2</font>,<font color="Maroon">y'</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x1</font>,<font color="Maroon">y'</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x2</font>,<font color="Maroon">x'</font></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i>, </i>;<br/>
<a NAME="E6:9_2_2_1_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> )
 
<b>by </b><i>, , <a class="ref" href="arytm_0.html#T12">ARYTM_0:12</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_2_2_1_1_1"/>
( ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) or (  <a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) or (  <a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) )
 </span><b>by </b><i>, , <a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_2_1_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:9_2_2_1_1_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 <b>and </b><br/><a NAME="E2:9_2_2_1_1_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E3:9_2_2_1_1_1_1"/>
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">y</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="arytm_0.html#D4">ARYTM_0:def 4</a></i>;<br/><a NAME="E4:9_2_2_1_1_1_1"/><b>then </b><i><font color="Green">E70</font></i>: 
 <a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D2" target="_self">Def2</a>, <a class="ref" href="arytm_0.html#T13">ARYTM_0:13</a></i>;<br/><a NAME="E5:9_2_2_1_1_1_1"/><i><font color="Green">E71</font></i>: 
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="arytm_0.html#T14">ARYTM_0:14</a></i>;<br/><a NAME="E6:9_2_2_1_1_1_1"/>
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="arytm_0.html#T14">ARYTM_0:14</a></i>;<br/><b>then </b><i><font color="Green">E72</font></i>: 1 = 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<font color="Maroon">y'</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> 0,<span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<font color="Maroon">x'</font></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i>, , , <a class="ref" href="arytm_0.html#T13">ARYTM_0:13</a></i>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<font color="Maroon">y'</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<font color="Maroon">x'</font></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="arytm_0.html#T13">ARYTM_0:13</a></i>
;<br/><a NAME="E8:9_2_2_1_1_1_1"/><i><font color="Green">E73</font></i>: 
1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">j</font>,0</span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i><a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>;<br/> <a href="arytm_0.html#K2">*</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">y'</font> = 
 <a href="arytm_0.html#K3">opp</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<font color="Maroon">y'</font></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T17">ARYTM_0:17</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="xcmplx_0.html#E3:9"><i><font color="Green">E20</font></i></a>, , <a class="ref" href="arytm_0.html#T12">ARYTM_0:12</a></i>
;<br/><a NAME="E10:9_2_2_1_1_1_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">y'</font> <a href="hidden.html#R1">=</a>  <a href="arytm_0.html#K4">inv</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="arytm_0.html#D5">ARYTM_0:def 5</a></i>;<br/><a NAME="E11:9_2_2_1_1_1_1"/><i><font color="Green">E75</font></i>: 
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="arytm_0.html#T14">ARYTM_0:14</a></i>;<br/> <a href="arytm_0.html#K2">*</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K4">inv</a> <font color="Maroon">y</font></span>)</span></span>)</span> = 
 <a href="arytm_0.html#K3">opp</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<span class="p2">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K4">inv</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T17">ARYTM_0:17</a></i>
<br/>.= 
 <a href="arytm_0.html#K3">opp</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4">inv</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T17">ARYTM_0:17</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="xcmplx_0.html#D2" target="_self">Def2</a>, <a class="ref" href="arytm_0.html#D5">ARYTM_0:def 5</a></i>
;<br/><a NAME="E13:9_2_2_1_1_1_1"/><b>then </b><i><font color="Green">E76</font></i>: 
 <a href="arytm_0.html#K4">inv</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="arytm_0.html#K3">opp</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K4">inv</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="arytm_0.html#D5">ARYTM_0:def 5</a></i>;<br/><a NAME="E14:9_2_2_1_1_1_1"/>
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="xcmplx_0.html#E3:9"><i><font color="Green">E20</font></i></a>, , <a class="ref" href="arytm_0.html#T12">ARYTM_0:12</a></i>;<br/><b>hence </b><font color="Maroon">z'</font> = 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default">0,<span class="p2">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i>, <a class="ref" href="xcmplx_0.html#D2" target="_self">Def2</a>, , <a class="ref" href="arytm_0.html#T23">ARYTM_0:23</a></i>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default">0,<span class="p2">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">j</font>,<span class="p4">(<span class="default"><a href="arytm_0.html#K4">inv</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i>, <a class="ref" href="arytm_0.html#T21">ARYTM_0:21</a></i>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default">0,<span class="p2">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<span class="p5">(<span class="default"><a href="arytm_0.html#K4">inv</a> <font color="Maroon">y</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K4">inv</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D2" target="_self">Def2</a>, <a class="ref" href="arytm_0.html#D5">ARYTM_0:def 5</a></i>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default">0,<span class="p2">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<span class="p4">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p5">(<span class="default"><a href="arytm_0.html#K4">inv</a> <font color="Maroon">y</font></span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K4">inv</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="arytm_0.html#T15">ARYTM_0:15</a></i>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default">0,<span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K4">inv</a> <font color="Maroon">y</font></span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K4">inv</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="arytm_0.html#T17">ARYTM_0:17</a></i>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default">0,<span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="arytm_0.html#T24">ARYTM_0:24</a></i>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default">0,<span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1">+</a> 0,<span class="p5">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="arytm_0.html#T13">ARYTM_0:13</a></i>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p5">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p5">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i>, , <a class="ref" href="arytm_0.html#T14">ARYTM_0:14</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_2_1_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:9_2_2_1_1_1_2"/><i><font color="Green">E77</font></i>: 
 <a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0
 <b>and </b><br/><a NAME="E2:9_2_2_1_1_1_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E3:9_2_2_1_1_1_2"/>
 <a href="arytm_0.html#K1">+</a> <font color="Maroon">y</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="arytm_0.html#D4">ARYTM_0:def 4</a></i>;<br/><a NAME="E4:9_2_2_1_1_1_2"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="arytm_0.html#T13">ARYTM_0:13</a></i>;<br/><a NAME="E5:9_2_2_1_1_1_2"/><b>then </b><i><font color="Green">E80</font></i>: 
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="arytm_0.html#T14">ARYTM_0:14</a></i>;<br/> <a href="arytm_0.html#K3">opp</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<font color="Maroon">y'</font></span>)</span> = 
 <a href="arytm_0.html#K2">*</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">y'</font>
<b>by </b><i><a class="ref" href="arytm_0.html#T17">ARYTM_0:17</a></i>
<br/>.= 
0
<b>by </b><i>, <a class="ref" href="arytm_0.html#T14">ARYTM_0:14</a></i>
;<br/><b>then </b><i><font color="Green">E81</font></i>: 1 = 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x'</font></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">y'</font></span>)</span>,0</span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i>, , , <a class="ref" href="arytm_0.html#T13">ARYTM_0:13</a></i>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x'</font></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">y'</font></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="arytm_0.html#T13">ARYTM_0:13</a></i>
;<br/><a NAME="E8:9_2_2_1_1_1_2"/><i><font color="Green">E84</font></i>: 
1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">j</font>,0</span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i><a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>;<br/><a NAME="E9:9_2_2_1_1_1_2"/><b>then </b>
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="xcmplx_0.html#E3:9"><i><font color="Green">E20</font></i></a>, <a class="ref" href="arytm_0.html#T12">ARYTM_0:12</a></i>;<br/><a NAME="E10:9_2_2_1_1_1_2"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a>  <a href="arytm_0.html#K4">inv</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D2" target="_self">Def2</a>, <a class="ref" href="arytm_0.html#D5">ARYTM_0:def 5</a></i>;<br/><a NAME="E11:9_2_2_1_1_1_2"/>
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="xcmplx_0.html#E3:9"><i><font color="Green">E20</font></i></a>, , <a class="ref" href="arytm_0.html#T12">ARYTM_0:12</a></i>;<br/><a NAME="E12:9_2_2_1_1_1_2"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D2" target="_self">Def2</a>, <a class="ref" href="arytm_0.html#T23">ARYTM_0:23</a></i>;<br/><a NAME="E13:9_2_2_1_1_1_2"/><i><font color="Green">E87</font></i>: 
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="arytm_0.html#T14">ARYTM_0:14</a></i>;<br/><font color="Maroon">x'</font> = 
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">j</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4">inv</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="arytm_0.html#T21">ARYTM_0:21</a></i>
<br/>.= 
 <a href="arytm_0.html#K2">*</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="arytm_0.html#K4">inv</a> <font color="Maroon">x</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4">inv</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D2" target="_self">Def2</a>, <a class="ref" href="arytm_0.html#D5">ARYTM_0:def 5</a></i>
<br/>.= 
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K4">inv</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K4">inv</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T15">ARYTM_0:15</a></i>
<br/>.= 
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T24">ARYTM_0:24</a></i>
<br/>.= 
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T13">ARYTM_0:13</a></i>
;<br/><b>hence </b><a NAME="E15:9_2_2_1_1_1_2"/>
<font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p5">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p5">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i>, , , , <a class="ref" href="arytm_0.html#T14">ARYTM_0:14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_2_1_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:9_2_2_1_1_1_3"/><i><font color="Green">E88</font></i>: 
 <a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E2:9_2_2_1_1_1_3"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><div><i><font color="Green">E90</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_2_1_1_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_2_2_1_1_1_3_1"/>
 <a href="arytm_0.html#K1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:9_2_2_1_1_1_3_1"/><b>then </b>
 <a href="arytm_0.html#K1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="arytm_0.html#T21">ARYTM_0:21</a></i>;<br/><a NAME="E3:9_2_2_1_1_1_3_1"/><b>then </b>
 <a href="arytm_0.html#K1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="arytm_0.html#D5">ARYTM_0:def 5</a></i>;<br/><a NAME="E4:9_2_2_1_1_1_3_1"/><b>then </b>
 <a href="arytm_0.html#K1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="arytm_0.html#T15">ARYTM_0:15</a></i>;<br/><a NAME="E5:9_2_2_1_1_1_3_1"/><b>then </b><i><font color="Green">E91</font></i>: 
 <a href="arytm_0.html#K2">*</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="arytm_0.html#T16">ARYTM_0:16</a></i>;<br/><a NAME="E6:9_2_2_1_1_1_3_1"/>
 <a href="arytm_0.html#K1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D2" target="_self">Def2</a>, <a class="ref" href="arytm_0.html#T19">ARYTM_0:19</a></i>;<br/><a NAME="E7:9_2_2_1_1_1_3_1"/><b>then </b><i><font color="Green">E92</font></i>: 
 <a href="arytm_0.html#K4">inv</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="arytm_0.html#T23">ARYTM_0:23</a></i>;<br/><a NAME="E8:9_2_2_1_1_1_3_1"/>
 <a href="arytm_0.html#K2">*</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="arytm_0.html#D5">ARYTM_0:def 5</a></i>;<br/><b>hence </b><a NAME="E9:9_2_2_1_1_1_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="xcmplx_0.html#E3:9"><i><font color="Green">E20</font></i></a>, <a class="ref" href="arytm_0.html#T14">ARYTM_0:14</a></i>;<br/></div><b>end;</b></div><b>reconsider </b><font color="Maroon">j</font> = 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/><a NAME="E5:9_2_2_1_1_1_3"/><i><font color="Green">E93</font></i>: 
1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><font color="Maroon">j</font>,0</span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i><a class="ref" href="arytm_0.html#D7">ARYTM_0:def 7</a></i>;<br/><a NAME="E6:9_2_2_1_1_1_3"/><b>then </b>
 <a href="arytm_0.html#K1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x1</font>,<font color="Maroon">y'</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x2</font>,<font color="Maroon">x'</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="arytm_0.html#T12">ARYTM_0:12</a></i>;<br/><a NAME="E7:9_2_2_1_1_1_3"/><b>then </b>
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">y'</font> <a href="hidden.html#R1">=</a>  <a href="arytm_0.html#K3">opp</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<font color="Maroon">x'</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="arytm_0.html#D4">ARYTM_0:def 4</a></i>;<br/><a NAME="E8:9_2_2_1_1_1_3"/><b>then </b>
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">y'</font> <a href="hidden.html#R1">=</a>  <a href="arytm_0.html#K2">*</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">x'</font>
 <b>by </b><i><a class="ref" href="arytm_0.html#T17">ARYTM_0:17</a></i>;<br/><b>then </b><i><font color="Green">E94</font></i>: <font color="Maroon">x'</font> = 
 <a href="arytm_0.html#K2">*</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">y'</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="arytm_0.html#T22">ARYTM_0:22</a></i>
<br/>.= 
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y'</font>,<span class="p2">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T15">ARYTM_0:15</a></i>
;<br/><a NAME="E10:9_2_2_1_1_1_3"/><b>then </b>
 <a href="arytm_0.html#K1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y'</font>,<span class="p4">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p5">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<font color="Maroon">y'</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, , , <a class="ref" href="arytm_0.html#T12">ARYTM_0:12</a></i>;<br/><a NAME="E11:9_2_2_1_1_1_3"/><b>then </b>
 <a href="arytm_0.html#K1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y'</font>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<font color="Maroon">y'</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="arytm_0.html#T15">ARYTM_0:15</a></i>;<br/><a NAME="E12:9_2_2_1_1_1_3"/><b>then </b>
 <a href="arytm_0.html#K1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y'</font>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">y'</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="arytm_0.html#T17">ARYTM_0:17</a></i>;<br/><a NAME="E13:9_2_2_1_1_1_3"/><b>then </b>
 <a href="arytm_0.html#K1">+</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y'</font>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">y'</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="arytm_0.html#T15">ARYTM_0:15</a></i>;<br/><a NAME="E14:9_2_2_1_1_1_3"/><b>then </b>
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">y'</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="arytm_0.html#T16">ARYTM_0:16</a></i>;<br/><a NAME="E15:9_2_2_1_1_1_3"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">y'</font> <a href="hidden.html#R1">=</a>  <a href="arytm_0.html#K4">inv</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="arytm_0.html#D5">ARYTM_0:def 5</a></i>;<br/><b>then </b><i><font color="Green">E96</font></i>: <font color="Maroon">x'</font> = 
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p5">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p0">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="xcmplx_0.html#E3:9"><i><font color="Green">E20</font></i></a>, <a class="ref" href="arytm_0.html#T24">ARYTM_0:24</a></i>
<br/>.= 
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p5">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p0">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T16">ARYTM_0:16</a></i>
<br/>.= 
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p5">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p0">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<span class="p5">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T17">ARYTM_0:17</a></i>
<br/>.= 
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p5">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p0">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p5">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T17">ARYTM_0:17</a></i>
<br/>.= 
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p5">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p0">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T15">ARYTM_0:15</a></i>
<br/>.= 
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<font color="Maroon">j</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="arytm_0.html#D5">ARYTM_0:def 5</a></i>
<br/>.= 
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T21">ARYTM_0:21</a></i>
;<br/><font color="Maroon">y'</font> = 
 <a href="arytm_0.html#K2">*</a> <font color="Maroon">j</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p5">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="arytm_0.html#T21">ARYTM_0:21</a></i>
<br/>.= 
 <a href="arytm_0.html#K2">*</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p5">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="arytm_0.html#D5">ARYTM_0:def 5</a></i>
<br/>.= 
 <a href="arytm_0.html#K2">*</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p5">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p0">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T15">ARYTM_0:15</a></i>
<br/>.= 
 <a href="arytm_0.html#K2">*</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p5">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p0">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T24">ARYTM_0:24</a></i>
<br/>.= 
 <a href="arytm_0.html#K2">*</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p5">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p0">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T16">ARYTM_0:16</a></i>
<br/>.= 
 <a href="arytm_0.html#K2">*</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p5">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p0">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T15">ARYTM_0:15</a></i>
<br/>.= 
 <a href="arytm_0.html#K2">*</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<font color="Maroon">j</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="arytm_0.html#D5">ARYTM_0:def 5</a></i>
<br/>.= 
 <a href="arytm_0.html#K2">*</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T21">ARYTM_0:21</a></i>
<br/>.= 
 <a href="arytm_0.html#K2">*</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<span class="p5">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T17">ARYTM_0:17</a></i>
<br/>.= 
 <a href="arytm_0.html#K2">*</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K3">opp</a> <span class="p5">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="arytm_0.html#T17">ARYTM_0:17</a></i>
<br/>.= 
 <a href="arytm_0.html#K2">*</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>

;<br/><b>hence </b><a NAME="E18:9_2_2_1_1_1_3"/>
<font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p5">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p5">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><font color="Maroon">c1</font> = 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p5">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2">*</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K3">opp</a> <font color="Maroon">y</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1">+</a> <span class="p5">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2">*</a> <font color="Maroon">y</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">c2</font>
<b>by </b><i>, </i>
;<br/><br/>


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

<b>thus </b><a NAME="E2:9_2_2"/>
( not <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c1</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c2</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">c1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c2</font> )
 ;<br/>


</div>
