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<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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1"/>
( <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 or <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="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1"><b>suppose </b></a><a NAME="E1:30_1_1"/>
<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 class="add"><b>hence </b><a NAME="E2:30_1_1"/>
( <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="xxreal_0.html#NR4" target="_self">&gt;</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">a</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> or <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><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2"><b>suppose </b></a><a NAME="E1:30_1_2"/><i><font color="Green">E31</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">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_2_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span><b>by </b><i><a class="ref" href="cqc_the1.html#T2">CQC_THE1:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_1_1"><b>suppose </b></a><a NAME="E1:30_1_2_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:30_1_2_1_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> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <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>
 <b>by </b><i/>;<br/><a NAME="E3:30_1_2_1_1"/>
<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>hence </b><a NAME="E4:30_1_2_1_1"/>
( <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="xxreal_0.html#NR4" target="_self">&gt;</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">a</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> or <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 )
 <b>by </b><i>, , , <a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_1_2"><b>suppose </b></a><a NAME="E1:30_1_2_1_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:30_1_2_1_2"/>
<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">0 <a href="nat_1.html#K1">+</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="funct_2.html#K3">.</a> 0</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">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:30_1_2_1_2"/>
( <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="xxreal_0.html#NR4" target="_self">&gt;</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">a</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> or <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 )
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_1_3"><b>suppose </b></a><a NAME="E1:30_1_2_1_3"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font> <a href="xcmplx_0.html#K6">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i/>;<br/><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">a</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> = 
<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">a1</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>

<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">a1</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="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E4:30_1_2_1_3"/>
( <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="xxreal_0.html#NR4" target="_self">&gt;</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">a</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> or <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 )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

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