<?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">G</font>;<br/>





<b>assume </b><a NAME="E1:25"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>
 ;<br/>

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

<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:25"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="int_1.html#T8">INT_1:8</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">k</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1"><b>suppose </b></a><a NAME="E1:25_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><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="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>;<br/><a NAME="E2:25_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:25_1_1_1"/>
<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">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="E2:25_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 <b>by </b><i><a class="ref" href="group_2.html#T55">GROUP_2:55</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:25_1_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">n</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">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_2"><b>proof </b></a><div class="add">

<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>assume </b><a NAME="E1:25_1_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<a NAME="E2:25_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">n</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>
 
<b>by </b><i>, , <a class="ref" href="group_2.html#T59">GROUP_2:59</a></i>;<br/>
<b>hence </b><a NAME="E3:25_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="ref" href="group_1.html#T49">GROUP_1:49</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:25_1_1"/>
for <font color="Olive">n</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">n</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="E5:25_1_1"/>
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">j</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_2"><b>suppose </b></a><a NAME="E1:25_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><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> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>;<br/><a NAME="E2:25_1_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:25_1_2_1"/>
<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">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="E2:25_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 <b>by </b><i><a class="ref" href="group_2.html#T55">GROUP_2:55</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:25_1_2"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">n</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">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2_2"><b>proof </b></a><div class="add">

<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>assume </b><a NAME="E1:25_1_2_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<a NAME="E2:25_1_2_2"/>
<font color="Maroon">a</font> <a href="group_1.html#K3">"</a>  <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>
 
<b>by </b><i>, <a class="ref" href="group_2.html#T60">GROUP_2:60</a></i>;<br/>
<a NAME="E3:25_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>
 
<b>by </b><i>, <a class="ref" href="group_2.html#T59">GROUP_2:59</a></i>;<br/>
<a NAME="E4:25_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="group_1.html#T62">GROUP_1:62</a></i>;<br/>
<a NAME="E5:25_1_2_2"/><b>then </b>
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="group_1.html#T63">GROUP_1:63</a></i>;<br/>
<b>hence </b><a NAME="E6:25_1_2_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div><a NAME="E4:25_1_2"/>
for <font color="Olive">n</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">n</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="E5:25_1_2"/>
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">j</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
