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

<b>let </b><font color="Maroon" title="c1">n</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:25"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/>

<b>set </b><font color="Maroon" title="c2">G</font> =  <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Segm</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">addint</a> <font color="Maroon" title="c1">n</font></span>)</span> #);<br/>
<a NAME="E2:25"/><i><font color="Green" title="E24">A2</font></i>: 
for <font color="Olive" title="b1">h</font>, <font color="Olive" title="b2">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Segm</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">addint</a> <font color="Maroon" title="c1">n</font></span>)</span> #)<br/> for <font color="Olive" title="b3">A</font>, <font color="Olive" title="b4">B</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Segm</a> <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">A</font> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">B</font> holds <br/><font color="Olive" title="b1">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">A</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Olive" title="b4">B</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="gr_cy_1.html#E1:25"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="gr_cy_1.html#D5" target="_self" title="GR_CY_1:def.5">Def5</a></i>;<br/>
<b>thus </b><a NAME="E3:25"/>
for <font color="Olive" title="b1">h</font>, <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">f</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Segm</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">addint</a> <font color="Maroon" title="c1">n</font></span>)</span> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">f</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">h</font>, <font color="Maroon" title="c4">g</font>, <font color="Maroon" title="c5">f</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Segm</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">addint</a> <font color="Maroon" title="c1">n</font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon" title="c6">A</font> = <font color="Maroon" title="c3">h</font>, <font color="Maroon" title="c7">B</font> = <font color="Maroon" title="c4">g</font>, <font color="Maroon" title="c8">C</font> = <font color="Maroon" title="c5">f</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Segm</a> <font color="Maroon" title="c1">n</font> ;<br/>
<a NAME="E2:25_1"/><i><font color="Green" title="E25">A3</font></i>: 
( <font color="Maroon" title="c6">A</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c7">B</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c8">C</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
;<br/>
<a NAME="E3:25_1"/><i><font color="Green" title="E26">A4</font></i>: 
<font color="Maroon" title="c3">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c7">B</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="gr_cy_1.html#E2:25"><i><font color="Green" title="E24">A2</font></i></a></i>;<br/>
<a NAME="E4:25_1"/><i><font color="Green" title="E27">A5</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c8">C</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="gr_cy_1.html#E2:25"><i><font color="Green" title="E24">A2</font></i></a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">f</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c7">B</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c8">C</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E2:25"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="gr_cy_1.html#E3:25_1"><i><font color="Green" title="E26">A4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c7">B</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c8">C</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E2:25_1"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="nat_d.html#T22" title="NAT_D:th.22">NAT_D:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c8">C</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">B</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c8">C</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E2:25_1"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="nat_d.html#T22" title="NAT_D:th.22">NAT_D:22</a></i>
<br/>.= 
<font color="Maroon" title="c3">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">f</font></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E2:25"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="gr_cy_1.html#E4:25_1"><i><font color="Green" title="E27">A5</font></i></a></i>
;<br/>


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

<b>reconsider </b><font color="Maroon" title="c3">e</font> = 0 as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Segm</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">addint</a> <font color="Maroon" title="c1">n</font></span>)</span> #) <b>by </b><i><a class="txt" href="gr_cy_1.html#E1:25"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="gr_cy_1.html#T10" target="_self" title="GR_CY_1:th.10">Th10</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">e</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a><br/>

<b>let </b><font color="Maroon" title="c4">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Segm</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">addint</a> <font color="Maroon" title="c1">n</font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c4">h</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Segm</a> <font color="Maroon" title="c1">n</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c6">A</font> = <font color="Maroon" title="c5">A</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E7:25"/><i><font color="Green" title="E25">A6</font></i>: 
<font color="Maroon" title="c6">A</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="gr_cy_1.html#E1:25"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="gr_cy_1.html#T10" target="_self" title="GR_CY_1:th.10">Th10</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c4">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c3">e</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E2:25"><i><font color="Green" title="E24">A2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">h</font>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E7:25"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="nat_d.html#T24" title="NAT_D:th.24">NAT_D:24</a></i>
;<br/>

<b>thus </b><font color="Maroon" title="c3">e</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">h</font> = 
<span class="p1">(<span class="default">0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c6">A</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E2:25"><i><font color="Green" title="E24">A2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">h</font>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E7:25"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="nat_d.html#T24" title="NAT_D:th.24">NAT_D:24</a></i>
;<br/>

<b>consider </b><font color="Maroon" title="c7">p</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E11:25"/><i><font color="Green" title="E26">A7</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">p</font>
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E7:25"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/>
<a NAME="E12:25"/><i><font color="Green" title="E27">A8</font></i>: 
<font color="Maroon" title="c7">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="gr_cy_1.html#E11:25"><i><font color="Green" title="E26">A7</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E13:25"/>
<font color="Maroon" title="c7">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="gr_cy_1.html#E1:25"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="nat_d.html#T1" title="NAT_D:th.1">NAT_D:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c7">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Segm</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" title="GR_CY_1:func.4">addint</a> <font color="Maroon" title="c1">n</font></span>)</span> #) <b>by </b><i><a class="ref" href="gr_cy_1.html#T10" target="_self" title="GR_CY_1:th.10">Th10</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">g</font>
;<br/>

<b>reconsider </b><font color="Maroon" title="c9">B</font> = <font color="Maroon" title="c8">g</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Segm</a> <font color="Maroon" title="c1">n</font> ;<br/>
<b>thus </b><a NAME="E16:25"/>
<font color="Maroon" title="c4">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_4_1_1"/>
( <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> or <font color="Maroon" title="c7">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> )
 </span><b>by </b><i><a class="txt" href="gr_cy_1.html#E12:25"><i><font color="Green" title="E27">A8</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_4_1_1_1"><b>suppose </b></a><a NAME="E1:25_4_1_1_1"/><i><font color="Green" title="E28">A9</font></i>: 
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 ;<br/><div class="add"><a NAME="E2:25_4_1_1_1"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E11:25"><i><font color="Green" title="E26">A7</font></i></a></i>;<br/><a NAME="E3:25_4_1_1_1"/><b>then </b><i><font color="Green" title="E29">A10</font></i>: 
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><font color="Maroon" title="c4">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">g</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">B</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E2:25"><i><font color="Green" title="E24">A2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E1:25_4_1_1_1"><i><font color="Green" title="E28">A9</font></i></a>, <a class="txt" href="gr_cy_1.html#E3:25_4_1_1_1"><i><font color="Green" title="E29">A10</font></i></a>, <a class="ref" href="nat_d.html#T25" title="NAT_D:th.25">NAT_D:25</a></i>
<br/>.= 
<font color="Maroon" title="c3">e</font>
<b>by </b><i><a class="ref" href="nat_d.html#T13" title="NAT_D:th.13">NAT_D:13</a></i>
;<br/><b>hence </b><a NAME="E5:25_4_1_1_1"/>
<font color="Maroon" title="c4">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_4_1_1_2"><b>suppose </b></a><a NAME="E1:25_4_1_1_2"/><i><font color="Green" title="E28">A11</font></i>: 
<font color="Maroon" title="c7">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font>
 ;<br/><div class="add"><font color="Maroon" title="c4">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">g</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">B</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E2:25"><i><font color="Green" title="E24">A2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E11:25"><i><font color="Green" title="E26">A7</font></i></a>, <a class="txt" href="gr_cy_1.html#E1:25_4_1_1_2"><i><font color="Green" title="E28">A11</font></i></a>, <a class="ref" href="nat_d.html#T24" title="NAT_D:th.24">NAT_D:24</a></i>
<br/>.= 
<font color="Maroon" title="c3">e</font>
<b>by </b><i><a class="ref" href="nat_d.html#T25" title="NAT_D:th.25">NAT_D:25</a></i>
;<br/><b>hence </b><a NAME="E3:25_4_1_1_2"/>
<font color="Maroon" title="c4">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:25_4"/>
<font color="Maroon" title="c4">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font>
 ;<br/>


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

<b>thus </b><a NAME="E17:25"/>
<font color="Maroon" title="c8">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_5_1_1"/>
( <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> or <font color="Maroon" title="c7">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> )
 </span><b>by </b><i><a class="txt" href="gr_cy_1.html#E12:25"><i><font color="Green" title="E27">A8</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_5_1_1_1"><b>suppose </b></a><a NAME="E1:25_5_1_1_1"/><i><font color="Green" title="E28">A12</font></i>: 
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 ;<br/><div class="add"><a NAME="E2:25_5_1_1_1"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">A</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E11:25"><i><font color="Green" title="E26">A7</font></i></a></i>;<br/><a NAME="E3:25_5_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><b>then </b><font color="Maroon" title="c8">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">h</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E2:25"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="gr_cy_1.html#E1:25_5_1_1_1"><i><font color="Green" title="E28">A12</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="nat_d.html#T25" title="NAT_D:th.25">NAT_D:25</a></i>
<br/>.= 
<font color="Maroon" title="c3">e</font>
<b>by </b><i><a class="ref" href="nat_d.html#T13" title="NAT_D:th.13">NAT_D:13</a></i>
;<br/><b>hence </b><a NAME="E5:25_5_1_1_1"/>
<font color="Maroon" title="c8">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_5_1_1_2"><b>suppose </b></a><a NAME="E1:25_5_1_1_2"/><i><font color="Green" title="E28">A13</font></i>: 
<font color="Maroon" title="c7">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font>
 ;<br/><div class="add"><font color="Maroon" title="c8">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">h</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">A</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c9">B</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E2:25"><i><font color="Green" title="E24">A2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E11:25"><i><font color="Green" title="E26">A7</font></i></a>, <a class="txt" href="gr_cy_1.html#E1:25_5_1_1_2"><i><font color="Green" title="E28">A13</font></i></a>, <a class="ref" href="nat_d.html#T24" title="NAT_D:th.24">NAT_D:24</a></i>
<br/>.= 
<font color="Maroon" title="c3">e</font>
<b>by </b><i><a class="ref" href="nat_d.html#T25" title="NAT_D:th.25">NAT_D:25</a></i>
;<br/><b>hence </b><a NAME="E3:25_5_1_1_2"/>
<font color="Maroon" title="c8">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:25_5"/>
<font color="Maroon" title="c8">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">e</font>
 ;<br/>


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


</div>
