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

<span class="kw">let </span><font color="Maroon" title="c5">c1</font>, <font color="Maroon" title="c6">c2</font> be   <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c2">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">z</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">z</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c5">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c2</font> ) &amp; (  not <font color="Maroon" title="c2">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c6">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c5">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c2</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E1:10_2_2"/>
( <font color="Maroon" title="c2">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">z</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">z</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c5">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c2</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c6">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c5">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c2</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:10_2_2_1"/><span class="lab"><font color="Green" title="E7">A19</font></span>: 
<font color="Maroon" title="c2">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E2:10_2_2_1"/><span class="lab"><font color="Green" title="E8">A20</font></span>: 
<font color="Maroon" title="c2">z</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">and </span><br/><a NAME="E3:10_2_2_1"/><span class="lab"><font color="Green" title="E9">A21</font></span>: 
<font color="Maroon" title="c2">z</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c2</font></span><br/>

<a NAME="E4:10_2_2_1"/><span class="lab"><font color="Green" title="E10">A22</font></span>: 
 for <font color="Olive" title="b1">z9</font> being   <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c2">z</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/><font color="Olive" title="b1">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">z9</font> be   <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">z</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c7">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:10_2_2_1_1"/><span class="lab"><font color="Green" title="E11">A23</font></span>: 
<font color="Maroon" title="c2">z</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c8">x1</font>, <font color="Maroon" title="c9">x2</font>, <font color="Maroon" title="c10">x9</font>, <font color="Maroon" title="c11">y9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E3:10_2_2_1_1"/><span class="lab"><font color="Green" title="E12">A24</font></span>: 
<font color="Maroon" title="c2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c9">x2</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <span class="kw">and </span><br/><a NAME="E4:10_2_2_1_1"/><span class="lab"><font color="Green" title="E13">A25</font></span>: 
<font color="Maroon" title="c7">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c10">x9</font>,<font color="Maroon" title="c11">y9</font></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <span class="kw">and </span><br/><a NAME="E5:10_2_2_1_1"/><span class="lab"><font color="Green" title="E14">A26</font></span>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c10">x9</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c11">y9</font>)</span>)</span></span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c11">y9</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c10">x9</font>)</span>)</span>)</span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_2_2_1_1"><span class="lab"><font color="Green" title="E11">A23</font></span></a>, <a class="ref" href="xcmplx_0.html#D5" target="_self" title="XCMPLX_0:def.5">Def5</a></span>;<br/>
<a NAME="E6:10_2_2_1_1"/><span class="lab"><font color="Green" title="E15">A27</font></span>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x1</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E3:10_2_2_1_1"><span class="lab"><font color="Green" title="E12">A24</font></span></a>, <a class="ref" href="arytm_0.html#T10" title="ARYTM_0:th.10">ARYTM_0:10</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_2_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:10_2_2_1_1_1"/>
( ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or (  <a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or (  <a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E1:10_2_2_1"><span class="lab"><font color="Green" title="E7">A19</font></span></a>, <a class="ref" href="arytm_0.html#D5" title="ARYTM_0:def.5">ARYTM_0:def 5</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_2_1_1_1_1"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:10_2_2_1_1_1_1"/><span class="lab"><font color="Green" title="E16">A28</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E2:10_2_2_1_1_1_1"/><span class="lab"><font color="Green" title="E17">A29</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span></span><br/><div class="add"><a NAME="E3:10_2_2_1_1_1_1"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<font color="Maroon" title="c4">y</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#D3" title="ARYTM_0:def.3">ARYTM_0:def 3</a></span>;<br/><a NAME="E4:10_2_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A30</font></span>: 
 <a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10_2_2_1_1_1_1"><span class="lab"><font color="Green" title="E17">A29</font></span></a>, <a class="ref" href="arytm_0.html#T11" title="ARYTM_0:th.11">ARYTM_0:11</a></span>;<br/><a NAME="E5:10_2_2_1_1_1_1"/>
(  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c11">y9</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c10">x9</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_2_2_1_1_1_1"><span class="lab"><font color="Green" title="E16">A28</font></span></a>, <a class="ref" href="arytm_0.html#T12" title="ARYTM_0:th.12">ARYTM_0:12</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A31</font></span>: 1 = 
<span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c11">y9</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c10">x9</font>)</span>)</span>)</span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10_2_2_1_1"><span class="lab"><font color="Green" title="E14">A26</font></span></a>, <a class="txt" href="#E6:10_2_2_1_1"><span class="lab"><font color="Green" title="E15">A27</font></span></a>, <a class="ref" href="arytm_0.html#T11" title="ARYTM_0:th.11">ARYTM_0:11</a></span>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c11">y9</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c10">x9</font>)</span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T11" title="ARYTM_0:th.11">ARYTM_0:11</a></span>
;<br/><a NAME="E7:10_2_2_1_1_1_1"/><span class="lab"><font color="Green" title="E20">A32</font></span>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c1">j</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#D5" title="ARYTM_0:def.5">ARYTM_0:def 5</a></span>;<br/> <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<font color="Maroon" title="c11">y9</font>) = 
 <a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c11">y9</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T15" title="ARYTM_0:th.15">ARYTM_0:15</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:10_2_2_1_1_1_1"><span class="lab"><font color="Green" title="E19">A31</font></span></a>, <a class="txt" href="#E7:10_2_2_1_1_1_1"><span class="lab"><font color="Green" title="E20">A32</font></span></a>, <a class="ref" href="arytm_0.html#T10" title="ARYTM_0:th.10">ARYTM_0:10</a></span>
;<br/><a NAME="E9:10_2_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A33</font></span>: 
<font color="Maroon" title="c11">y9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10_2_2_1_1_1_1"><span class="lab"><font color="Green" title="E18">A30</font></span></a>, <a class="ref" href="arytm_0.html#D4" title="ARYTM_0:def.4">ARYTM_0:def 4</a></span>;<br/><a NAME="E10:10_2_2_1_1_1_1"/><span class="lab"><font color="Green" title="E22">A34</font></span>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_2_2_1_1_1_1"><span class="lab"><font color="Green" title="E16">A28</font></span></a>, <a class="ref" href="arytm_0.html#T12" title="ARYTM_0:th.12">ARYTM_0:12</a></span>;<br/> <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>) = 
 <a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<span class="p2">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T15" title="ARYTM_0:th.15">ARYTM_0:15</a></span>
<br/>.= 
 <a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c4">y</font></span>)</span>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T15" title="ARYTM_0:th.15">ARYTM_0:15</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10_2_2_1_1_1_1"><span class="lab"><font color="Green" title="E17">A29</font></span></a>, <a class="ref" href="arytm_0.html#D4" title="ARYTM_0:def.4">ARYTM_0:def 4</a></span>
;<br/><a NAME="E12:10_2_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A35</font></span>: 
 <a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c4">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10_2_2_1_1_1_1"><span class="lab"><font color="Green" title="E18">A30</font></span></a>, <a class="ref" href="arytm_0.html#D4" title="ARYTM_0:def.4">ARYTM_0:def 4</a></span>;<br/><a NAME="E13:10_2_2_1_1_1_1"/>
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c10">x9</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:10_2_2_1_1_1_1"><span class="lab"><font color="Green" title="E19">A31</font></span></a>, <a class="txt" href="#E7:10_2_2_1_1_1_1"><span class="lab"><font color="Green" title="E20">A32</font></span></a>, <a class="ref" href="arytm_0.html#T10" title="ARYTM_0:th.10">ARYTM_0:10</a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c7">z9</font> = 
<span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<span class="p2">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10_2_2_1_1"><span class="lab"><font color="Green" title="E13">A25</font></span></a>, <a class="txt" href="#E2:10_2_2_1_1_1_1"><span class="lab"><font color="Green" title="E17">A29</font></span></a>, <a class="txt" href="#E9:10_2_2_1_1_1_1"><span class="lab"><font color="Green" title="E21">A33</font></span></a>, <a class="ref" href="arytm_0.html#T21" title="ARYTM_0:th.21">ARYTM_0:21</a></span>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<span class="p2">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c1">j</font>,<span class="p4">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c4">y</font></span>)</span>)</span>)</span></span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:10_2_2_1_1_1_1"><span class="lab"><font color="Green" title="E23">A35</font></span></a>, <a class="ref" href="arytm_0.html#T19" title="ARYTM_0:th.19">ARYTM_0:19</a></span>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<span class="p2">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<span class="p5">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c4">y</font></span>)</span>)</span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c4">y</font></span>)</span>)</span>)</span></span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10_2_2_1_1_1_1"><span class="lab"><font color="Green" title="E17">A29</font></span></a>, <a class="ref" href="arytm_0.html#D4" title="ARYTM_0:def.4">ARYTM_0:def 4</a></span>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<span class="p2">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c4">y</font></span>)</span>)</span>)</span>)</span>)</span></span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T13" title="ARYTM_0:th.13">ARYTM_0:13</a></span>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p4">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c4">y</font></span>)</span>)</span>)</span>)</span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T15" title="ARYTM_0:th.15">ARYTM_0:15</a></span>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span></span>)</span>)</span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T22" title="ARYTM_0:th.22">ARYTM_0:22</a></span>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T11" title="ARYTM_0:th.11">ARYTM_0:11</a></span>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_2_2_1_1_1_1"><span class="lab"><font color="Green" title="E16">A28</font></span></a>, <a class="txt" href="#E10:10_2_2_1_1_1_1"><span class="lab"><font color="Green" title="E22">A34</font></span></a>, <a class="ref" href="arytm_0.html#T12" title="ARYTM_0:th.12">ARYTM_0:12</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_2_1_1_1_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:10_2_2_1_1_1_2"/><span class="lab"><font color="Green" title="E16">A36</font></span>: 
 <a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E2:10_2_2_1_1_1_2"/><span class="lab"><font color="Green" title="E17">A37</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span></span><br/><div class="add"><a NAME="E3:10_2_2_1_1_1_2"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<font color="Maroon" title="c4">y</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#D3" title="ARYTM_0:def.3">ARYTM_0:def 3</a></span>;<br/><a NAME="E4:10_2_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A38</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_2_2_1_1_1_2"><span class="lab"><font color="Green" title="E16">A36</font></span></a>, <a class="ref" href="arytm_0.html#T11" title="ARYTM_0:th.11">ARYTM_0:11</a></span>;<br/><a NAME="E5:10_2_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A39</font></span>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c10">x9</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T12" title="ARYTM_0:th.12">ARYTM_0:12</a></span>;<br/> <a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c11">y9</font>)</span>)</span> = 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<font color="Maroon" title="c11">y9</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T15" title="ARYTM_0:th.15">ARYTM_0:15</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_2_2_1_1_1_2"><span class="lab"><font color="Green" title="E16">A36</font></span></a>, <a class="ref" href="arytm_0.html#T12" title="ARYTM_0:th.12">ARYTM_0:12</a></span>
;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A40</font></span>: 1 = 
<span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c10">x9</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c11">y9</font>)</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10_2_2_1_1"><span class="lab"><font color="Green" title="E14">A26</font></span></a>, <a class="txt" href="#E6:10_2_2_1_1"><span class="lab"><font color="Green" title="E15">A27</font></span></a>, <a class="txt" href="#E5:10_2_2_1_1_1_2"><span class="lab"><font color="Green" title="E19">A39</font></span></a>, <a class="ref" href="arytm_0.html#T11" title="ARYTM_0:th.11">ARYTM_0:11</a></span>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c10">x9</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c11">y9</font>)</span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T11" title="ARYTM_0:th.11">ARYTM_0:11</a></span>
;<br/><a NAME="E8:10_2_2_1_1_1_2"/><span class="lab"><font color="Green" title="E21">A41</font></span>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c1">j</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#D5" title="ARYTM_0:def.5">ARYTM_0:def 5</a></span>;<br/><a NAME="E9:10_2_2_1_1_1_2"/><span class="kw">then </span>
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c10">x9</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:10_2_2_1_1_1_2"><span class="lab"><font color="Green" title="E20">A40</font></span></a>, <a class="ref" href="arytm_0.html#T10" title="ARYTM_0:th.10">ARYTM_0:10</a></span>;<br/><a NAME="E10:10_2_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A42</font></span>: 
<font color="Maroon" title="c10">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10_2_2_1_1_1_2"><span class="lab"><font color="Green" title="E17">A37</font></span></a>, <a class="ref" href="arytm_0.html#D4" title="ARYTM_0:def.4">ARYTM_0:def 4</a></span>;<br/><a NAME="E11:10_2_2_1_1_1_2"/>
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c11">y9</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:10_2_2_1_1_1_2"><span class="lab"><font color="Green" title="E20">A40</font></span></a>, <a class="txt" href="#E8:10_2_2_1_1_1_2"><span class="lab"><font color="Green" title="E21">A41</font></span></a>, <a class="ref" href="arytm_0.html#T10" title="ARYTM_0:th.10">ARYTM_0:10</a></span>;<br/><a NAME="E12:10_2_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A43</font></span>: 
<font color="Maroon" title="c11">y9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10_2_2_1_1_1_2"><span class="lab"><font color="Green" title="E17">A37</font></span></a>, <a class="ref" href="arytm_0.html#T21" title="ARYTM_0:th.21">ARYTM_0:21</a></span>;<br/><a NAME="E13:10_2_2_1_1_1_2"/><span class="lab"><font color="Green" title="E24">A44</font></span>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10_2_2_1_1_1_2"><span class="lab"><font color="Green" title="E18">A38</font></span></a>, <a class="ref" href="arytm_0.html#T12" title="ARYTM_0:th.12">ARYTM_0:12</a></span>;<br/><font color="Maroon" title="c10">x9</font> = 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c1">j</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c3">x</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:10_2_2_1_1_1_2"><span class="lab"><font color="Green" title="E22">A42</font></span></a>, <a class="ref" href="arytm_0.html#T19" title="ARYTM_0:th.19">ARYTM_0:19</a></span>
<br/>.= 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c3">x</font></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c3">x</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10_2_2_1_1_1_2"><span class="lab"><font color="Green" title="E17">A37</font></span></a>, <a class="ref" href="arytm_0.html#D4" title="ARYTM_0:def.4">ARYTM_0:def 4</a></span>
<br/>.= 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p2">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c3">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <font color="Maroon" title="c3">x</font></span>)</span>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T13" title="ARYTM_0:th.13">ARYTM_0:13</a></span>
<br/>.= 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T22" title="ARYTM_0:th.22">ARYTM_0:22</a></span>
<br/>.= 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>)</span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T11" title="ARYTM_0:th.11">ARYTM_0:11</a></span>
;<br/><span class="kw">hence </span><a NAME="E15:10_2_2_1_1_1_2"/>
<font color="Maroon" title="c7">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10_2_2_1_1"><span class="lab"><font color="Green" title="E13">A25</font></span></a>, <a class="txt" href="#E1:10_2_2_1_1_1_2"><span class="lab"><font color="Green" title="E16">A36</font></span></a>, <a class="txt" href="#E12:10_2_2_1_1_1_2"><span class="lab"><font color="Green" title="E23">A43</font></span></a>, <a class="txt" href="#E13:10_2_2_1_1_1_2"><span class="lab"><font color="Green" title="E24">A44</font></span></a>, <a class="ref" href="arytm_0.html#T12" title="ARYTM_0:th.12">ARYTM_0:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_2_1_1_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:10_2_2_1_1_1_3"/><span class="lab"><font color="Green" title="E16">A45</font></span>: 
 <a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E2:10_2_2_1_1_1_3"/><span class="lab"><font color="Green" title="E17">A46</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span></span><br/><div class="add"><div><span class="lab"><font color="Green" title="E18">A47</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_2_1_1_1_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:10_2_2_1_1_1_3_1"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:10_2_2_1_1_1_3_1"/><span class="kw">then </span>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p2">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<font color="Maroon" title="c1">j</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T19" title="ARYTM_0:th.19">ARYTM_0:19</a></span>;<br/><a NAME="E3:10_2_2_1_1_1_3_1"/><span class="kw">then </span>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p2">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>)</span>)</span>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_2_2_1_1_1_3"><span class="lab"><font color="Green" title="E16">A45</font></span></a>, <a class="ref" href="arytm_0.html#D4" title="ARYTM_0:def.4">ARYTM_0:def 4</a></span>;<br/><a NAME="E4:10_2_2_1_1_1_3_1"/><span class="kw">then </span>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T13" title="ARYTM_0:th.13">ARYTM_0:13</a></span>;<br/><a NAME="E5:10_2_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A48</font></span>: 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>)</span>)</span>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T14" title="ARYTM_0:th.14">ARYTM_0:14</a></span>;<br/><a NAME="E6:10_2_2_1_1_1_3_1"/>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p2">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>)</span>)</span>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10_2_2_1_1_1_3"><span class="lab"><font color="Green" title="E17">A46</font></span></a>, <a class="ref" href="arytm_0.html#T17" title="ARYTM_0:th.17">ARYTM_0:17</a></span>;<br/><a NAME="E7:10_2_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A49</font></span>: 
 <a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10_2_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E19">A48</font></span></a>, <a class="ref" href="arytm_0.html#T21" title="ARYTM_0:th.21">ARYTM_0:21</a></span>;<br/><a NAME="E8:10_2_2_1_1_1_3_1"/>
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_2_2_1_1_1_3"><span class="lab"><font color="Green" title="E16">A45</font></span></a>, <a class="ref" href="arytm_0.html#D4" title="ARYTM_0:def.4">ARYTM_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E9:10_2_2_1_1_1_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:10_2_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E20">A49</font></span></a>, <a class="ref" href="arytm_0.html#T12" title="ARYTM_0:th.12">ARYTM_0:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">reconsider </span><font color="Maroon" title="c12">j</font> = 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/><a NAME="E5:10_2_2_1_1_1_3"/><span class="lab"><font color="Green" title="E19">A50</font></span>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><font color="Maroon" title="c12">j</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#D5" title="ARYTM_0:def.5">ARYTM_0:def 5</a></span>;<br/><a NAME="E6:10_2_2_1_1_1_3"/><span class="kw">then </span>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c11">y9</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c10">x9</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10_2_2_1_1"><span class="lab"><font color="Green" title="E14">A26</font></span></a>, <a class="ref" href="arytm_0.html#T10" title="ARYTM_0:th.10">ARYTM_0:10</a></span>;<br/><a NAME="E7:10_2_2_1_1_1_3"/><span class="kw">then </span>
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c11">y9</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c10">x9</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:10_2_2_1_1"><span class="lab"><font color="Green" title="E15">A27</font></span></a>, <a class="ref" href="arytm_0.html#D3" title="ARYTM_0:def.3">ARYTM_0:def 3</a></span>;<br/><a NAME="E8:10_2_2_1_1_1_3"/><span class="kw">then </span>
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c11">y9</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<font color="Maroon" title="c10">x9</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T15" title="ARYTM_0:th.15">ARYTM_0:15</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A51</font></span>: <font color="Maroon" title="c10">x9</font> = 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c11">y9</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_2_2_1_1_1_3"><span class="lab"><font color="Green" title="E16">A45</font></span></a>, <a class="ref" href="arytm_0.html#T20" title="ARYTM_0:th.20">ARYTM_0:20</a></span>
<br/>.= 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c11">y9</font>,<span class="p2">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T13" title="ARYTM_0:th.13">ARYTM_0:13</a></span>
;<br/><a NAME="E10:10_2_2_1_1_1_3"/><span class="kw">then </span>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c11">y9</font>,<span class="p4">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p5">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>)</span>)</span>)</span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c11">y9</font>)</span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10_2_2_1_1"><span class="lab"><font color="Green" title="E14">A26</font></span></a>, <a class="txt" href="#E6:10_2_2_1_1"><span class="lab"><font color="Green" title="E15">A27</font></span></a>, <a class="txt" href="#E5:10_2_2_1_1_1_3"><span class="lab"><font color="Green" title="E19">A50</font></span></a>, <a class="ref" href="arytm_0.html#T10" title="ARYTM_0:th.10">ARYTM_0:10</a></span>;<br/><a NAME="E11:10_2_2_1_1_1_3"/><span class="kw">then </span>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c11">y9</font>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>)</span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c11">y9</font>)</span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T13" title="ARYTM_0:th.13">ARYTM_0:13</a></span>;<br/><a NAME="E12:10_2_2_1_1_1_3"/><span class="kw">then </span>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c11">y9</font>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>)</span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p2">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<font color="Maroon" title="c11">y9</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T15" title="ARYTM_0:th.15">ARYTM_0:15</a></span>;<br/><a NAME="E13:10_2_2_1_1_1_3"/><span class="kw">then </span>
 <a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c11">y9</font>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>)</span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p2">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<font color="Maroon" title="c11">y9</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T13" title="ARYTM_0:th.13">ARYTM_0:13</a></span>;<br/><a NAME="E14:10_2_2_1_1_1_3"/><span class="kw">then </span>
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c11">y9</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T14" title="ARYTM_0:th.14">ARYTM_0:14</a></span>;<br/><a NAME="E15:10_2_2_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A52</font></span>: 
<font color="Maroon" title="c11">y9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p1">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10_2_2_1_1_1_3"><span class="lab"><font color="Green" title="E18">A47</font></span></a>, <a class="ref" href="arytm_0.html#D4" title="ARYTM_0:def.4">ARYTM_0:def 4</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A53</font></span>: <font color="Maroon" title="c10">x9</font> = 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p0">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>)</span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>)</span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>)</span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:10_2_2_1_1_1_3"><span class="lab"><font color="Green" title="E20">A51</font></span></a>, <a class="ref" href="arytm_0.html#T22" title="ARYTM_0:th.22">ARYTM_0:22</a></span>
<br/>.= 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p0">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>)</span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>)</span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p4">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>)</span>)</span>)</span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T14" title="ARYTM_0:th.14">ARYTM_0:14</a></span>
<br/>.= 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p0">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>)</span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>)</span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<span class="p5">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>)</span>)</span></span>)</span>)</span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T15" title="ARYTM_0:th.15">ARYTM_0:15</a></span>
<br/>.= 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p0">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>)</span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>)</span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T15" title="ARYTM_0:th.15">ARYTM_0:15</a></span>
<br/>.= 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p0">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>)</span>)</span>)</span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T13" title="ARYTM_0:th.13">ARYTM_0:13</a></span>
<br/>.= 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<font color="Maroon" title="c12">j</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_2_2_1_1_1_3"><span class="lab"><font color="Green" title="E16">A45</font></span></a>, <a class="ref" href="arytm_0.html#D4" title="ARYTM_0:def.4">ARYTM_0:def 4</a></span>
<br/>.= 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T19" title="ARYTM_0:th.19">ARYTM_0:19</a></span>
;<br/><font color="Maroon" title="c11">y9</font> = 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c12">j</font>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p5">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>)</span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>)</span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:10_2_2_1_1_1_3"><span class="lab"><font color="Green" title="E21">A52</font></span></a>, <a class="ref" href="arytm_0.html#T19" title="ARYTM_0:th.19">ARYTM_0:19</a></span>
<br/>.= 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p2">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p5">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>)</span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>)</span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_2_2_1_1_1_3"><span class="lab"><font color="Green" title="E16">A45</font></span></a>, <a class="ref" href="arytm_0.html#D4" title="ARYTM_0:def.4">ARYTM_0:def 4</a></span>
<br/>.= 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p2">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p3">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p0">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>)</span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>)</span>)</span></span>)</span>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T13" title="ARYTM_0:th.13">ARYTM_0:13</a></span>
<br/>.= 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p0">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>)</span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>)</span>)</span>)</span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T22" title="ARYTM_0:th.22">ARYTM_0:22</a></span>
<br/>.= 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p4">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p0">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>)</span>)</span>)</span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p4">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>)</span>)</span>)</span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T14" title="ARYTM_0:th.14">ARYTM_0:14</a></span>
<br/>.= 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p0">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span></span>)</span>)</span>)</span>)</span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p4">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>)</span>)</span>)</span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T13" title="ARYTM_0:th.13">ARYTM_0:13</a></span>
<br/>.= 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<font color="Maroon" title="c12">j</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p4">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>)</span>)</span>)</span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10_2_2_1_1_1_3"><span class="lab"><font color="Green" title="E16">A45</font></span></a>, <a class="ref" href="arytm_0.html#D4" title="ARYTM_0:def.4">ARYTM_0:def 4</a></span>
<br/>.= 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p4">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p4">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>)</span>)</span>)</span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T19" title="ARYTM_0:th.19">ARYTM_0:19</a></span>
<br/>.= 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<span class="p5">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>)</span>)</span></span>)</span>)</span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T15" title="ARYTM_0:th.15">ARYTM_0:15</a></span>
<br/>.= 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#T15" title="ARYTM_0:th.15">ARYTM_0:15</a></span>
<br/>.= 
 <a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p1">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p2">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)

;<br/><span class="kw">hence </span><a NAME="E18:10_2_2_1_1_1_3"/>
<font color="Maroon" title="c7">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10_2_2_1_1"><span class="lab"><font color="Green" title="E13">A25</font></span></a>, <a class="txt" href="#E16:10_2_2_1_1_1_3"><span class="lab"><font color="Green" title="E22">A53</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><font color="Maroon" title="c5">c1</font> = 
<span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<span class="p3">(<span class="default"><a href="arytm_0.html#K3" title="ARYTM_0:func.3">opp</a> <font color="Maroon" title="c4">y</font></span>)</span>,<span class="p3">(<span class="default"><a href="arytm_0.html#K4" title="ARYTM_0:func.4">inv</a> <span class="p4">(<span class="default"><a href="arytm_0.html#K1" title="ARYTM_0:func.1">+</a> (<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>)</span>)</span>,<span class="p5">(<span class="default"><a href="arytm_0.html#K2" title="ARYTM_0:func.2">*</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>)</span>)</span></span>)</span>)</span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10_2_2_1"><span class="lab"><font color="Green" title="E8">A20</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c6">c2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10_2_2_1"><span class="lab"><font color="Green" title="E9">A21</font></span></a>, <a class="txt" href="#E4:10_2_2_1"><span class="lab"><font color="Green" title="E10">A22</font></span></a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E2:10_2_2"/>
(  not <font color="Maroon" title="c2">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c6">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c5">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
