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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</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>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>3</sub></font> <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> <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">a<sub>1</sub></font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> );<br/>
<a NAME="E1:49"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:49_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>thus </b><font color="Maroon">c<sub>3</sub></font> <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> = 
<font color="Maroon">c<sub>3</sub></font> <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>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E50"><i><font color="Green">Lemma29</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D5" target="_self">Def5</a></i>
<br/>.= 
<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> <font color="Maroon">c<sub>3</sub></font>
<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> 0</span>)</span> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font>
<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="E2:49"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</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="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:49_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>3</sub></font> <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> <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>5</sub></font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>assume </b><a NAME="E2:49_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>thus </b><font color="Maroon">c<sub>3</sub></font> <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="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<font color="Maroon">c<sub>3</sub></font> <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#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#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#K1" target="_self">*</a> <font color="Maroon">c<sub>4</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>
<b>by </b><i><a class="ref" href="group_1.html#D4" target="_self">Def4</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <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> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E1:49_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="group_1.html#E2:49_2"><i><font color="Green">E41</font></i></a>, <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#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> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font>
<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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#T49" target="_self">Th49</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E3:49"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</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="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="group_1.html#E1:49"><i><font color="Green">E38</font></i></a>, <a class="txt" href="group_1.html#E2:49"><i><font color="Green">E39</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:49"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>3</sub></font> <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>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="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>


</div>
