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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>







<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="int_1.html#NM1">Integer</a>] means for <font color="Olive">b<sub>1</sub></font> being  <a href="int_1.html#NM1">Integer</a> holds <font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Olive">b<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>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E1:60"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/>

<b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K2">+</a> 0</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">c<sub>3</sub></font></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>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E50"><i><font color="Green">Lemma29</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E2:60"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> 1] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/>

<b>assume </b><a NAME="E1:60_2"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="int_1.html#NM1">Integer</a> holds <font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Olive">b<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>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E2:60_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="int_1.html#NM1">Integer</a> holds <font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Olive">b<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>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/>

<b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> = 
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="xcmplx_0.html#K6">-</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>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E1:60_2"><i><font color="Green">E52</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</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></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E78"><i><font color="Green">Lemma46</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>6</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>4</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>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>5</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"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K2">+</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#E1:60_2"><i><font color="Green">E52</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span>

;<br/>


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

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/>

<b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> = 
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</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>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E1:60_2"><i><font color="Green">E52</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</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>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E80"><i><font color="Green">Lemma48</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>6</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>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> 1</span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>5</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"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E1:60_2"><i><font color="Green">E52</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E3:60"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="int_1.html#NM1">Integer</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="int_1.html#S4">INT_1:sch 4</a>(<a class="txt" href="group_1.html#E1:60"><i><font color="Green">E50</font></i></a>, <a class="txt" href="group_1.html#E2:60"><i><font color="Green">E51</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:60"/>
<font color="Maroon">c<sub>4</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> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</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>4</sub></font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>


</div>
