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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<div><i><font color="Green">E49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="binop_2.html#K7">-</a> 1 or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> 1 )
 </span><b>by </b><i><a class="txt" href="group_1.html#E79"><i><font color="Green">Lemma47</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1"><b>suppose </b></a><a NAME="E1:59_1_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E3:59_1_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><a NAME="E4:59_1_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="int_2.html#K1">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E3:59_1_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="group_1.html#T55" target="_self">Th55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3" target="_self">"</a> </span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T62" target="_self">Th62</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3" target="_self">"</a> </span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K3" target="_self">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E76"><i><font color="Green">Lemma44</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3" target="_self">"</a> </span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p4">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="group_1.html#K3" target="_self">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E1:59_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="group_1.html#T55" target="_self">Th55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p4">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_1.html#K3" target="_self">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T25" target="_self">Th25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K3" target="_self">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T49" target="_self">Th49</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p3">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K3" target="_self">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E4:59_1_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
 <a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D6" target="_self">Def6</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_2"><b>suppose </b></a><a NAME="E1:59_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="binop_2.html#K7">-</a> 1
 ;<br/><div class="add"><a NAME="E2:59_1_1_2"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p3">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K3" target="_self">"</a> </span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T56" target="_self">Th56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p3">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K3" target="_self">"</a> </span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3" target="_self">"</a> </span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T62" target="_self">Th62</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p4">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K3" target="_self">"</a> </span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3" target="_self">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4" target="_self">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p4">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K3" target="_self">"</a> </span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T25" target="_self">Th25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K3" target="_self">"</a> </span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T49" target="_self">Th49</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K3" target="_self">"</a> </span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E2:59_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
 <a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D6" target="_self">Def6</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_3"><b>suppose </b></a><a NAME="E1:59_1_1_3"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> 1
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E50"><i><font color="Green">Lemma29</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D5" target="_self">Def5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3" target="_self">"</a> </span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T62" target="_self">Th62</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3" target="_self">"</a> </span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K3" target="_self">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E76"><i><font color="Green">Lemma44</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3" target="_self">"</a> </span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K3" target="_self">"</a> </span>)</span> <a href="group_1.html#K3" target="_self">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E1:59_1_1_3"><i><font color="Green">E50</font></i></a>, <a class="ref" href="group_1.html#T62" target="_self">Th62</a></i>
<br/>.= 
 <a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D6" target="_self">Def6</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K3" target="_self">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E76"><i><font color="Green">Lemma44</font></i></a></i>
<br/>.= 
 <a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D6" target="_self">Def6</a></i>
;<br/>
<a NAME="E3:59"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="group_1.html#E1:59"><i><font color="Green">E49</font></i></a>, <a class="ref" href="group_1.html#T14" target="_self">Th14</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> = 
<font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K3" target="_self">"</a> 
<b>by </b><i><a class="txt" href="group_1.html#E3:59"><i><font color="Green">E50</font></i></a>, <a class="txt" href="group_1.html#E76"><i><font color="Green">Lemma44</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="group_1.html#K3" target="_self">"</a> </span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span></span>)</span> <a href="group_1.html#K3" target="_self">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T25" target="_self">Th25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span></span>)</span> <a href="group_1.html#K3" target="_self">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E76"><i><font color="Green">Lemma44</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> <span class="p3">(<span class="default"><a href="binop_2.html#K7">-</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E76"><i><font color="Green">Lemma44</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6" target="_self">|^</a> 1</span>)</span>

;<br/>


</div>
