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

<b>let </b><font color="Maroon">n</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:22"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>set </b><font color="Maroon">G</font> =  <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="gr_cy_1.html#K1" target="_self">Segm</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" target="_self">addint</a> <font color="Maroon">n</font></span>)</span> #);<br/>
<a NAME="E2:22"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">h</font>, <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="gr_cy_1.html#K1" target="_self">Segm</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" target="_self">addint</a> <font color="Maroon">n</font></span>)</span> #)<br/> for <font color="Olive">A</font>, <font color="Olive">B</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1" target="_self">Segm</a> <font color="Maroon">n</font>  st <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> &amp; <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> holds <br/><font color="Olive">h</font> <a href="group_1.html#K1">*</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">A</font> <a href="nat_1.html#K1">+</a> <font color="Olive">B</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
 
<b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E3:22"/>
for <font color="Olive">h</font>, <font color="Olive">g</font>, <font color="Olive">f</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="gr_cy_1.html#K1" target="_self">Segm</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" target="_self">addint</a> <font color="Maroon">n</font></span>)</span> #) holds  <span class="p1">(<span class="default"><font color="Olive">h</font> <a href="group_1.html#K1">*</a> <font color="Olive">g</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">g</font> <a href="group_1.html#K1">*</a> <font color="Olive">f</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">h</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="gr_cy_1.html#K1" target="_self">Segm</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" target="_self">addint</a> <font color="Maroon">n</font></span>)</span> #), <font color="Maroon">g</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="gr_cy_1.html#K1" target="_self">Segm</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" target="_self">addint</a> <font color="Maroon">n</font></span>)</span> #), <font color="Maroon">f</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="gr_cy_1.html#K1" target="_self">Segm</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" target="_self">addint</a> <font color="Maroon">n</font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">h</font>, <font color="Maroon">B</font> = <font color="Maroon">g</font>, <font color="Maroon">C</font> = <font color="Maroon">f</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1" target="_self">Segm</a> <font color="Maroon">n</font> ;<br/>
<a NAME="E2:22_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">A</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">B</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">C</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  )
 
;<br/>
<a NAME="E3:22_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">B</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:22_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">C</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">B</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">C</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">B</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">C</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i>, <a class="ref" href="nat_1.html#T74">NAT_1:74</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">C</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i>, <a class="ref" href="nat_1.html#T74">NAT_1:74</a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i>, </i>
;<br/>


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

<b>reconsider </b><font color="Maroon">e</font> = 0 as   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="gr_cy_1.html#K1" target="_self">Segm</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" target="_self">addint</a> <font color="Maroon">n</font></span>)</span> #) <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">e</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D3">GROUP_1:def 3</a><br/>

<b>let </b><font color="Maroon">h</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="gr_cy_1.html#K1" target="_self">Segm</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" target="_self">addint</a> <font color="Maroon">n</font></span>)</span> #);<br/>

<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">h</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1" target="_self">Segm</a> <font color="Maroon">n</font> ;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E7:22"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">A</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 
<b>by </b><i>, </i>;<br/>
<b>thus </b><font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">e</font> = 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="nat_1.html#K1">+</a> 0</span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">h</font>
<b>by </b><i>, <a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>
;<br/>

<b>thus </b><font color="Maroon">e</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> = 
<span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> <font color="Maroon">A</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">h</font>
<b>by </b><i>, <a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>
;<br/>

<b>consider </b><font color="Maroon">p</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E11:22"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<a NAME="E12:22"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E13:22"/>
<font color="Maroon">p</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">p</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="gr_cy_1.html#K1" target="_self">Segm</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" target="_self">addint</a> <font color="Maroon">n</font></span>)</span> #) <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">g</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1" target="_self">Segm</a> <font color="Maroon">n</font> ;<br/>
<b>thus </b><a NAME="E16:22"/>
<font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_4_1_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> or <font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_1_1_1"><b>suppose </b></a><a NAME="E1:22_4_1_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:22_4_1_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E3:22_4_1_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g</font> = 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">B</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i>, , <a class="ref" href="nat_1.html#T77">NAT_1:77</a></i>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="nat_1.html#T63">NAT_1:63</a></i>
;<br/><b>hence </b><a NAME="E5:22_4_1_1_1"/>
<font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_4_1_1_2"><b>suppose </b></a><a NAME="E1:22_4_1_1_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g</font> = 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">B</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i>, , <a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="nat_1.html#T77">NAT_1:77</a></i>
;<br/><b>hence </b><a NAME="E3:22_4_1_1_2"/>
<font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:22_4"/>
<font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 ;<br/>


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

<b>thus </b><a NAME="E17:22"/>
<font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_5_1_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> or <font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_5_1_1_1"><b>suppose </b></a><a NAME="E1:22_5_1_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:22_5_1_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E3:22_5_1_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>then </b><font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 0</span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="nat_1.html#T77">NAT_1:77</a></i>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="nat_1.html#T63">NAT_1:63</a></i>
;<br/><b>hence </b><a NAME="E5:22_5_1_1_1"/>
<font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_5_1_1_2"><b>suppose </b></a><a NAME="E1:22_5_1_1_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> = 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">B</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i>, , <a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="nat_1.html#T77">NAT_1:77</a></i>
;<br/><b>hence </b><a NAME="E3:22_5_1_1_2"/>
<font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:22_5"/>
<font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 ;<br/>


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


</div>
