<?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">a</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>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>( <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> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0 implies <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="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 );<br/>
<a NAME="E1:32"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_1"/>
<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> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:32_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<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> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><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/>
<a NAME="E3:32_1"/><i><font color="Green">E33</font></i>: 
<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> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<b>assume </b><a NAME="E4:32_1"/><i><font color="Green">E38</font></i>: 
<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> 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E5:32_1"/><b>then </b><i><font color="Green">E48</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>
 
<b>by </b><i/>;<br/>
<a NAME="E6:32_1"/><b>then </b>
<font color="Maroon">n</font> <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/>
<a NAME="E7:32_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">n</font> <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>
 
<b>by </b><i>, , , <a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/>
<a NAME="E8:32_1"/>
<font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:32_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>
<a NAME="E10:32_1"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 
;<br/>
<b>hence </b><a NAME="E11:32_1"/>
contradiction
 <b>by </b><i><a class="txt" href="real_3.html#E1"><i><font color="Green">Lemma28</font></i></a>, <a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:32"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">b</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>proof </b></a><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="E1:32_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">b</font>]
 ;<br/>

<b>assume </b><a NAME="E2:32_2"/>
<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> <span class="p1">(<span class="default"><span class="p2">(<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</span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E3:32_2"/><b>then </b>
<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> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
;<br/>
<a NAME="E4:32_2"/><b>then </b><i><font color="Green">E57</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="seq_1.html#K2">.</a> <font color="Maroon">b</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="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> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">x</font> = <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="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>;<br/>
<b>assume </b><a NAME="E5:32_2"/><i><font color="Green">E58</font></i>: 
<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="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E6:32_2"/><b>then </b><i><font color="Green">E59</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="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E7:32_2"/><b>then </b>
<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="seq_1.html#K2">.</a> <font color="Maroon">b</font> <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="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#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="seq_1.html#K2">.</a> <font color="Maroon">b</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="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</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="seq_1.html#K2">.</a> <font color="Maroon">b</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="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="nat_1.html#T47">NAT_1:47</a></i>;<br/>
<a NAME="E8:32_2"/><b>then </b><i><font color="Green">E68</font></i>: 
<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="seq_1.html#K2">.</a> <font color="Maroon">b</font> <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="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/>
<a NAME="E9:32_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E10:32_2"/><b>then </b>
<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="seq_1.html#K2">.</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E11:32_2"/>
contradiction
 <b>by </b><i><a class="txt" href="real_3.html#E1"><i><font color="Green">Lemma28</font></i></a>, <a class="txt" href="real_3.html#E2"><i><font color="Green">Lemma29</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:32"/>
for <font color="Olive">b</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );<br/></i>
<b>hence </b><a NAME="E4:32"/>
( <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> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> 0 implies <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="seq_1.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>


</div>
