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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:41"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:41"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="gr_cy_1.html#K5" target="_self">INT.Group</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <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">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" target="_self">addint</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="gr_cy_1.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E3:41"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K5" target="_self">INT.Group</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></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">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="gr_cy_1.html#E1:41"><i><font color="Green">E35</font></i></a>, <a class="txt" href="gr_cy_1.html#E2:41"><i><font color="Green">E36</font></i></a>, <a class="ref" href="gr_cy_1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E4:41"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="gr_cy_1.html#E1:41"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E5:41"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span><b>by </b><i><a class="txt" href="gr_cy_1.html#E5:41"><i><font color="Green">E38</font></i></a>, <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="41_1_1_1"><b>suppose </b></a><a NAME="E1:41_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>then </b>the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K5" target="_self">INT.Group</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p3">(<span class="default"><a href="gr_cy_1.html#K5" target="_self">INT.Group</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E2:41"><i><font color="Green">E36</font></i></a>, <a class="ref" href="gr_cy_1.html#T13" target="_self">Th13</a>, <a class="ref" href="gr_cy_1.html#T36" target="_self">Th36</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_2.html#K5">(1).</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K5" target="_self">INT.Group</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_2.html#D7">GROUP_2:def 7</a></i>
;<br/><a NAME="E3:41_1_1_1"/><b>then </b>
 <a href="gr_cy_1.html#K5" target="_self">INT.Group</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_2.html#K5">(1).</a> <span class="p1">(<span class="default"><a href="gr_cy_1.html#K5" target="_self">INT.Group</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="group_2.html#T70">GROUP_2:70</a></i>;<br/><a NAME="E4:41_1_1_1"/><b>then </b>
 <a href="gr_cy_1.html#K5" target="_self">INT.Group</a> <font color="Maroon">c<sub>1</sub></font> is  <a href="gr_cy_1.html#V1" target="_self">cyclic</a> <a href="group_1.html#NM1">Group</a>
 ;<br/><b>hence </b><a NAME="E5:41_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K5" target="_self">INT.Group</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K5" target="_self">INT.Group</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>ex <font color="Olive">b<sub>3</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K6">|^</a> <font color="Olive">b<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="gr_cy_1.html#T41" target="_self">Th41</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_2"><b>suppose </b></a><a NAME="E1:41_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1" target="_self">Segm</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="gr_cy_1.html#T10" target="_self">Th10</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K5" target="_self">INT.Group</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="gr_cy_1.html#E2:41"><i><font color="Green">E36</font></i></a></i>;<br/><a NAME="E4:41_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K5" target="_self">INT.Group</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>ex <font color="Olive">b<sub>2</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <font color="Olive">b<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K5" target="_self">INT.Group</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="gr_cy_1.html#E1:41"><i><font color="Green">E35</font></i></a>, <a class="txt" href="gr_cy_1.html#E2:41"><i><font color="Green">E36</font></i></a>, <a class="txt" href="gr_cy_1.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">c<sub>3</sub></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">a<sub>1</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> 0 = 
 <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="gr_cy_1.html#K5" target="_self">INT.Group</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T43">GROUP_1:43</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="gr_cy_1.html#E1:41"><i><font color="Green">E35</font></i></a>, <a class="ref" href="gr_cy_1.html#T36" target="_self">Th36</a></i>
<br/>.= 
0 <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="nat_1.html#T78">NAT_1:78</a></i>
;<br/>
<a NAME="E3:41_1_1_2_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E4:41_1_1_2_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:41_1_1_2_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="gr_cy_1.html#E1:41"><i><font color="Green">E35</font></i></a>, <a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="gr_cy_1.html#T10" target="_self">Th10</a></i>;<br/>
<b>assume </b><a NAME="E3:41_1_1_2_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T48">GROUP_1:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#T44">GROUP_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E3:41"><i><font color="Green">E37</font></i></a>, <a class="txt" href="gr_cy_1.html#E3:41_1_1_2_1_2"><i><font color="Green">E41</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="nat_1.html#T74">NAT_1:74</a></i>
;<br/>
<b>hence </b><a NAME="E5:41_1_1_2_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:41_1_1_2_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="gr_cy_1.html#E3:41_1_1_2_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="gr_cy_1.html#E4:41_1_1_2_1"><i><font color="Green">E40</font></i></a>);<br/></i>
<a NAME="E6:41_1_1_2_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="gr_cy_1.html#E1:41"><i><font color="Green">E35</font></i></a>, <a class="txt" href="gr_cy_1.html#E2:41"><i><font color="Green">E36</font></i></a>, <a class="ref" href="gr_cy_1.html#T10" target="_self">Th10</a></i>;<br/>
<i><font color="Green">E43</font></i>: <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E5:41_1_1_2_1"><i><font color="Green">E41</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E6:41_1_1_2_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>thus </b><a NAME="E9:41_1_1_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E7:41_1_1_2_1"><i><font color="Green">E43</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:41_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K5" target="_self">INT.Group</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K5" target="_self">INT.Group</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>ex <font color="Olive">b<sub>3</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K6">|^</a> <font color="Olive">b<sub>3</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:41"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K5" target="_self">INT.Group</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K5" target="_self">INT.Group</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>ex <font color="Olive">b<sub>3</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K6">|^</a> <font color="Olive">b<sub>3</sub></font>
 ;<br/>


</div>
