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

<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/><b>let </b><font color="Maroon" title="c2">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/><b>let </b><font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>







<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c2">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> $1;<br/>
<a NAME="E1:44"/><i><font color="Green" title="E36">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>thus </b><font color="Maroon" title="c2">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> = 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font>
<b>by </b><i><a class="ref" href="group_1.html#D8" target="_self" title="GROUP_1:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="group_1.html#D8" target="_self" title="GROUP_1:def.8">Def8</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E2:44"/><i><font color="Green" title="E37">A2</font></i>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:44_2"/><i><font color="Green" title="E38">A3</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c2">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c5">m</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c6">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>thus </b><font color="Maroon" title="c2">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> = 
<font color="Maroon" title="c2">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> 1</span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">m</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c6">n</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T48" target="_self" title="GROUP_1:th.48">Th48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c6">n</font></span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E1:44_2"><i><font color="Green" title="E38">A3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T44" target="_self" title="GROUP_1:th.44">Th44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T48" target="_self" title="GROUP_1:th.48">Th48</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E3:44"/>
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="group_1.html#E1:44"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="group_1.html#E2:44"><i><font color="Green" title="E37">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:44"/>
<font color="Maroon" title="c2">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c4">m</font>
 ;<br/>


</div>
