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

<b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>





<b>set </b><font color="Maroon">fd</font> =  <a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font>;<br/>
<b>set </b><font color="Maroon">fm</font> =  <a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font>;<br/>
<b>set </b><font color="Maroon">r</font> = <font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font>;<br/>
<i><font color="Green">E31</font></i>: <span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 0 = 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">m</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">n</font>
<b>by </b><i/>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:63_2"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_1"><b>suppose </b></a><a NAME="E1:63_2_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:63_2_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T49">XCMPLX_1:49</a></i>;<br/><a NAME="E3:63_2_1"/><i><font color="Green">E38</font></i>: 
 <a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> 0
 <b>by </b><i><a class="txt" href="real_3.html#E5:63_2_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="real_3.html#E1"><i><font color="Green">Lemma28</font></i></a></i>;<br/><a NAME="E4:63_2_1"/><i><font color="Green">E48</font></i>: 
 <a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <a href="numbers.html#K5">NAT</a>  <a href="funcop_1.html#K2">--&gt;</a> 0
 <b>by </b><i><a class="txt" href="real_3.html#E5:63_2_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="real_3.html#E2"><i><font color="Green">Lemma29</font></i></a></i>;<br/><a NAME="E5:63_2_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E6:63_2_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or  ex <font color="Olive">x</font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="xcmplx_0.html#K2">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> = 
0
<b>by </b><i>, <a class="ref" href="real_3.html#T4" target="_self">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="real_3.html#T9" target="_self">Th9</a>, <a class="ref" href="real_3.html#T10" target="_self">Th10</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><br/><b>thus </b><span class="p1">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 1 = 
<span class="p1">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span>

<br/>.= 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="real_3.html#E5:63_2_1"><i><font color="Green">E49</font></i></a>, , <a class="ref" href="real_3.html#T3" target="_self">Th3</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> = 
<span class="p1">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>

<br/>.= 
0 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="real_3.html#T3" target="_self">Th3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="real_3.html#T10" target="_self">Th10</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_2"><b>suppose </b></a><a NAME="E1:63_2_2"/><i><font color="Green">E50</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:63_2_2"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T47">NAT_1:47</a></i>;<br/><a NAME="E3:63_2_2"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T114">XCMPLX_1:114</a></i>;<br/><i><font color="Green">E58</font></i>: <span class="p1">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="int_1.html#K4">frac</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="txt" href="real_3.html#E1:63_2_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xcmplx_1.html#T57">XCMPLX_1:57</a></i>
<br/>.= 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span>
<b>by </b><i/>
;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>( ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> ) &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> ) );<br/><a NAME="E5:63_2_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_2_2"><b>proof </b></a><div class="add">

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:63_2_2_2_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><a NAME="E2:63_2_2_2_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><a NAME="E3:63_2_2_2_1"/>
<span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div>

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

<b>assume </b><a NAME="E2:63_2_2_2"/>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/>

<b>hence </b><a NAME="E3:63_2_2_2"/>
<span class="p1">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div><a NAME="E6:63_2_2"/><i><font color="Green">E68</font></i>: 
for <font color="Olive">a</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">a</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">a</font> <a href="xcmplx_0.html#K2">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_2_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:63_2_2_3"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">a</font>]
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:63_2_2_3_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_2_3_1_1"><b>suppose </b></a><a NAME="E1:63_2_2_3_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><i><font color="Green">E77</font></i>: <span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="int_1.html#K4">frac</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 0
<b>by </b><i><a class="ref" href="real_3.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><a href="int_1.html#K4">frac</a> <span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><a href="int_1.html#K4">frac</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T57">XCMPLX_1:57</a></i>
<br/>.= 
<font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> 1
<b>by </b><i/>
;<br/><div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:63_2_2_3_1_1_2"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><a NAME="E2:63_2_2_3_1_1_2"/><b>then </b>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, <a class="ref" href="cqc_the1.html#T2">CQC_THE1:2</a></i>;<br/><b>hence </b><a NAME="E3:63_2_2_3_1_1_2"/>
<span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="real_3.html#D2" target="_self">Def2</a></i>;<br/></div>
<b>end;</b></div><b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E4:63_2_2_3_1_1"/>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><a NAME="E5:63_2_2_3_1_1"/><b>then </b><i><font color="Green">E109</font></i>: 
( <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, <a class="ref" href="cqc_the1.html#T2">CQC_THE1:2</a></i>;<br/><b>set </b><font color="Maroon">x</font> = <font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_2_3_1_1_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:63_2_2_3_1_1_3"/>
( <font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_2_3_1_1_3_1"><b>suppose </b></a><a NAME="E1:63_2_2_3_1_1_3_1"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>thus </b><span class="p1">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> = 
<span class="p1">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="real_3.html#E3:63_2_2"><i><font color="Green">E57</font></i></a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="int_1.html#K4">frac</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="real_3.html#T9" target="_self">Th9</a>, <a class="ref" href="real_3.html#D2" target="_self">Def2</a>, </i>
<br/>.= 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">0 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="real_3.html#E4:63_2_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xcmplx_1.html#T49">XCMPLX_1:49</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">0 <a href="real_1.html#K5">-</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="real_3.html#E4:63_2_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="nat_1.html#D1">NAT_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K6">/</a> 0

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="real_3.html#E4:63_2_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="real_3.html#E3:63_2_2"><i><font color="Green">E57</font></i></a>, </i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_2_3_1_1_3_2"><b>suppose </b></a><a NAME="E1:63_2_2_3_1_1_3_2"/><i><font color="Green">E111</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:63_2_2_3_1_1_3_2"/><b>then </b><i><font color="Green">E112</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T47">NAT_1:47</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_2_3_1_1_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:63_2_2_3_1_1_3_2_1"/>
( <font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_2_3_1_1_3_2_1_1"><b>suppose </b></a><a NAME="E1:63_2_2_3_1_1_3_2_1_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:63_2_2_3_1_1_3_2_1_1"/><b>then </b><i><font color="Green">E114</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> = 
<span class="p1">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="real_3.html#E3:63_2_2"><i><font color="Green">E57</font></i></a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="int_1.html#K4">frac</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="real_3.html#T9" target="_self">Th9</a>, <a class="ref" href="real_3.html#D2" target="_self">Def2</a>, </i>
<br/>.= 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
1 <a href="real_1.html#K6">/</a> 0
<b>by </b><i><a class="ref" href="real_3.html#T12" target="_self">Th12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="real_3.html#D3" target="_self">Def3</a>, <a class="ref" href="xcmplx_1.html#T49">XCMPLX_1:49</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="real_3.html#E3:63_2_2"><i><font color="Green">E57</font></i></a>, </i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_2_3_1_1_3_2_1_2"><b>suppose </b></a><a NAME="E1:63_2_2_3_1_1_3_2_1_2"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:63_2_2_3_1_1_3_2_1_2"/><b>then </b><i><font color="Green">E116</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i/>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> = 
<span class="p1">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="real_3.html#E3:63_2_2"><i><font color="Green">E57</font></i></a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="int_1.html#K4">frac</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="real_3.html#T9" target="_self">Th9</a>, <a class="ref" href="real_3.html#D2" target="_self">Def2</a>, </i>
<br/>.= 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
1 <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="real_3.html#E5:63_2_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="real_3.html#E4:63_2_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="real_3.html#T12" target="_self">Th12</a>, <a class="ref" href="real_3.html#D3" target="_self">Def3</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="real_3.html#E3:63_2_2"><i><font color="Green">E57</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="real_3.html#E3:63_2_2"><i><font color="Green">E57</font></i></a>, </i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_2_3_1_2"><b>suppose </b></a><a NAME="E1:63_2_2_3_1_2"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:63_2_2_3_1_2"/><b>then </b><i><font color="Green">E117</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">b</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:63_2_2_3_1_2_1"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><a NAME="E2:63_2_2_3_1_2_1"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> 1 or <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> 1 )
 <b>by </b><i><a class="txt" href="real_3.html#E3:63_2_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_2_3_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:63_2_2_3_1_2_1_1"/>
( <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> or <font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 </span><b>by </b><i><a class="txt" href="real_3.html#E4:63_2_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_2_3_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:63_2_2_3_1_2_1_1_1"/>
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:63_2_2_3_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="real_3.html#T11" target="_self">Th11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_2_3_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:63_2_2_3_1_2_1_1_2"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">a2</font> = <font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E3:63_2_2_3_1_2_1_1_2"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i><a class="txt" href="real_3.html#E5:63_2_2"><i><font color="Green">E59</font></i></a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">b</font> = 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">b</font></span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a2</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">a2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
<b>by </b><i><a class="ref" href="real_3.html#D3" target="_self">Def3</a>, <a class="ref" href="real_3.html#T11" target="_self">Th11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a2</font></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="real_3.html#D3" target="_self">Def3</a>, </i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div><b>let </b><font color="Maroon">b</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E4:63_2_2_3_1_2"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_2_3_1_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:63_2_2_3_1_2_2"/>
( <font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> 1 or <font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="real_3.html#E3:63_2_2"><i><font color="Green">E57</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="63_2_2_3_1_2_2_1"><b>suppose </b></a><a NAME="E1:63_2_2_3_1_2_2_1"/>
<font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><div class="add"><a NAME="E2:63_2_2_3_1_2_2_1"/><b>then </b>
<font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E3:63_2_2_3_1_2_2_1"/>
<span class="p1">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="real_3.html#T11" target="_self">Th11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_2_3_1_2_2_2"><b>suppose </b></a><a NAME="E1:63_2_2_3_1_2_2_2"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">b1</font> = <font color="Maroon">b</font> <a href="xcmplx_0.html#K6">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E3:63_2_2_3_1_2_2_2"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><a NAME="E4:63_2_2_3_1_2_2_2"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">b1</font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E5:63_2_2_3_1_2_2_2"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_2_3_1_2_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:63_2_2_3_1_2_2_2_1"/>
( 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> or 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_2_3_1_2_2_2_1_1"><b>suppose </b></a><a NAME="E1:63_2_2_3_1_2_2_2_1_1"/><i><font color="Green">E130</font></i>: 
0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><b>thus </b><span class="p1">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> = 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="real_3.html#T12" target="_self">Th12</a>, <a class="ref" href="real_3.html#T1" target="_self">Th1</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="real_3.html#E4:63_2_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="real_3.html#D2" target="_self">Def2</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">b1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="real_3.html#E5:63_2_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="real_3.html#E4:63_2_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="real_3.html#T11" target="_self">Th11</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font></span>)</span> <a href="real_1.html#K6">/</a> 0</span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font></span>)</span> <a href="nat_1.html#K5">div</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="real_3.html#T13" target="_self">Th13</a>, <a class="ref" href="real_3.html#D3" target="_self">Def3</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="real_3.html#T13" target="_self">Th13</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_2_3_1_2_2_2_1_2"><b>suppose </b></a><a NAME="E1:63_2_2_3_1_2_2_2_1_2"/><i><font color="Green">E131</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:63_2_2_3_1_2_2_2_1_2"/><b>then </b><i><font color="Green">E132</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font></span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">b1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T47">NAT_1:47</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_2_3_1_2_2_2_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:63_2_2_3_1_2_2_2_1_2_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_2_3_1_2_2_2_1_2_1_1"><b>suppose </b></a><a NAME="E1:63_2_2_3_1_2_2_2_1_2_1_1"/><i><font color="Green">E133</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:63_2_2_3_1_2_2_2_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:63_2_2_3_1_2_2_2_1_2_1_1"/><b>then </b><i><font color="Green">E134</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>thus </b><span class="p1">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> = 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="real_3.html#T12" target="_self">Th12</a>, <a class="ref" href="real_3.html#T1" target="_self">Th1</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="real_3.html#E4:63_2_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="real_3.html#D2" target="_self">Def2</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">b1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="real_3.html#E5:63_2_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="real_3.html#E4:63_2_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="real_3.html#T11" target="_self">Th11</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> 0
<b>by </b><i>, <a class="ref" href="real_3.html#D3" target="_self">Def3</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T49">XCMPLX_1:49</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="real_3.html#D3" target="_self">Def3</a>, </i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_2_2_3_1_2_2_2_1_2_1_2"><b>suppose </b></a><a NAME="E1:63_2_2_3_1_2_2_2_1_2_1_2"/><i><font color="Green">E135</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:63_2_2_3_1_2_2_2_1_2_1_2"/><b>then </b><i><font color="Green">E136</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i/>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> = 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="real_3.html#T12" target="_self">Th12</a>, <a class="ref" href="real_3.html#T1" target="_self">Th1</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="real_3.html#E4:63_2_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="real_3.html#D2" target="_self">Def2</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">b1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="real_3.html#E5:63_2_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="real_3.html#E4:63_2_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="real_3.html#T11" target="_self">Th11</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">b1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">b1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="real_3.html#D3" target="_self">Def3</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">b1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="real_3.html#T14" target="_self">Th14</a>, , <a class="ref" href="real_3.html#T13" target="_self">Th13</a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="real_3.html#D3" target="_self">Def3</a>, </i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E7:63_2_2"/>
for <font color="Olive">a</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">a</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, <a class="ref" href="real_3.html#T10" target="_self">Th10</a>);<br/></i><b>hence </b><a NAME="E8:63_2_2"/>
( <span class="p1">(<span class="default"><a href="real_3.html#NK6" target="_self">scf</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_3.html#K2" target="_self">divSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> &amp; <span class="p1">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span> &amp; <span class="p1">(<span class="default"><a href="real_3.html#NK4" target="_self">rfs</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_3.html#K1" target="_self">modSeq</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="real_3.html#T9" target="_self">Th9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
