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

<b>let </b><font color="Maroon">G</font> be  <a href="group_1.html#V1">strict</a> <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">H</font> be  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/>







<b>assume </b><a NAME="E1:3"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/>

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E2:3"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>proof </b></a><div class="add">

<a NAME="E1:3_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> 0 <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="group_1.html#T43">GROUP_1:43</a></i>;<br/>
<a NAME="E2:3_1"/>
<font color="Maroon">b</font> <a href="group_1.html#K6">|^</a> 0 <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="group_1.html#T43">GROUP_1:43</a></i>;<br/>
<b>hence </b><a NAME="E3:3_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 <b>by </b><i>, <a class="ref" href="group_2.html#T53">GROUP_2:53</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:3"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:3_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<a NAME="E2:3_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="group_1.html#T49">GROUP_1:49</a></i>;<br/>
<a NAME="E3:3_2"/>
<font color="Maroon">b</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="group_1.html#T49">GROUP_1:49</a></i>;<br/>
<b>hence </b><a NAME="E4:3_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:3"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E5:3"/>
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">n</font>
 ;<br/>


</div>
