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<b>let </b><font color="Maroon" title="c1">q</font>, <font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:3"/><i><font color="Green" title="E4">A1</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">a</font>
 <b>and </b><br/><a NAME="E2:3"/><i><font color="Green" title="E5">A2</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">q</font>
 <b>and </b><br/><a NAME="E3:3"/><i><font color="Green" title="E6">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 ;<br/>

<a NAME="E4:3"/><i><font color="Green" title="E7">A4</font></i>: 
0 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">q</font>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E2:3"><i><font color="Green" title="E5">A2</font></i></a></i>;<br/>
<a NAME="E5:3"/><b>then </b><i><font color="Green" title="E8">A5</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">q</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">r</font> = <font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font>;<br/>
<b>set </b><font color="Maroon" title="c5">qNa</font> = <font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font>;<br/>
<b>set </b><font color="Maroon" title="c6">qNr</font> = <font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> $1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1;<br/>
<a NAME="E6:3"/><i><font color="Green" title="E9">A6</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="weddwitt.html#E1:3"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></i>;<br/>
<a NAME="E7:3"/><b>then </b><i><font color="Green" title="E10">A7</font></i>: 
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>by </b><i><a class="txt" href="weddwitt.html#E3:3"><i><font color="Green" title="E6">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E9:3"/><i><font color="Green" title="E11">A8</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">k</font>]
 <b>and </b><br/><a NAME="E10:3"/><i><font color="Green" title="E12">A9</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="weddwitt.html#E7:3"><i><font color="Green" title="E10">A7</font></i></a>);<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:3_1_1"/>
( <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c7">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_1"><b>suppose </b></a><a NAME="E1:3_1_1_1"/><i><font color="Green" title="E13">A10</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><div><i><font color="Green" title="E14">A11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:3_1_1_1_1"/><i><font color="Green" title="E15">A12</font></i>: 
0 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 ;<br/><a NAME="E2:3_1_1_1_1"/><b>then </b><i><font color="Green" title="E16">A13</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></i>;<br/><a NAME="E3:3_1_1_1_1"/>
<font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="weddwitt.html#E1:3_1_1_1_1"><i><font color="Green" title="E15">A12</font></i></a>, <a class="ref" href="nat_2.html#T10" title="NAT_2:th.10">NAT_2:10</a></i>;<br/><a NAME="E4:3_1_1_1_1"/><b>then </b><i><font color="Green" title="E17">A14</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></i>;<br/><a NAME="E5:3_1_1_1_1"/>
<font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E1:3"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="nat_d.html#T1" title="NAT_D:th.1">NAT_D:1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E7:3_1_1_1_1"/><i><font color="Green" title="E18">A15</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><a NAME="E8:3_1_1_1_1"/><i><font color="Green" title="E19">A16</font></i>: 
<font color="Maroon" title="c8">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="weddwitt.html#E1:3"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="weddwitt.html#E7:3_1_1_1_1"><i><font color="Green" title="E18">A15</font></i></a>, <a class="ref" href="nat_d.html#T1" title="NAT_D:th.1">NAT_D:1</a></i>;<br/><a NAME="E9:3_1_1_1_1"/><i><font color="Green" title="E20">A17</font></i>: 
<font color="Maroon" title="c1">q</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">q</font> <a href="prepower.html#K7" title="PREPOWER:func.7">#Z</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="prepower.html#T46" title="PREPOWER:th.46">PREPOWER:46</a></i>;<br/><a NAME="E10:3_1_1_1_1"/><i><font color="Green" title="E21">A18</font></i>: 
<font color="Maroon" title="c1">q</font> <a href="prepower.html#K7" title="PREPOWER:func.7">#Z</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="prepower.html#K7" title="PREPOWER:func.7">#Z</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="prepower.html#K7" title="PREPOWER:func.7">#Z</a> <font color="Maroon" title="c8">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E4:3"><i><font color="Green" title="E7">A4</font></i></a>, <a class="ref" href="prepower.html#T54" title="PREPOWER:th.54">PREPOWER:54</a></i>;<br/><a NAME="E11:3_1_1_1_1"/><i><font color="Green" title="E22">A19</font></i>: 
<font color="Maroon" title="c1">q</font> <a href="prepower.html#K7" title="PREPOWER:func.7">#Z</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="prepower.html#T46" title="PREPOWER:th.46">PREPOWER:46</a></i>;<br/><a NAME="E12:3_1_1_1_1"/><i><font color="Green" title="E23">A20</font></i>: 
<font color="Maroon" title="c1">q</font> <a href="prepower.html#K7" title="PREPOWER:func.7">#Z</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">q</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c8">m</font>
 <b>by </b><i><a class="ref" href="prepower.html#T46" title="PREPOWER:th.46">PREPOWER:46</a></i>;<br/><a NAME="E13:3_1_1_1_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">q</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E4:3"><i><font color="Green" title="E7">A4</font></i></a>, <a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></i>;<br/><a NAME="E14:3_1_1_1_1"/><b>then </b><i><font color="Green" title="E24">A21</font></i>: 
<font color="Maroon" title="c1">q</font> <a href="prepower.html#K7" title="PREPOWER:func.7">#Z</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="ref" href="prepower.html#T49" title="PREPOWER:th.49">PREPOWER:49</a></i>;<br/><a NAME="E15:3_1_1_1_1"/><i><font color="Green" title="E25">A22</font></i>: 
<font color="Maroon" title="c1">q</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="weddwitt.html#E2:3"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="prepower.html#T19" title="PREPOWER:th.19">PREPOWER:19</a></i>;<br/><a NAME="E16:3_1_1_1_1"/>
<font color="Maroon" title="c8">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E17:3_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c1">q</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <b>by </b><i><a class="txt" href="weddwitt.html#E2:3"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="weddwitt.html#E8:3_1_1_1_1"><i><font color="Green" title="E19">A16</font></i></a>, <a class="ref" href="weddwitt.html#T1" target="_self" title="WEDDWITT:th.1">Th1</a></i>;<br/><a NAME="E18:3_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c1">q</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>by </b><i><a class="txt" href="weddwitt.html#E15:3_1_1_1_1"><i><font color="Green" title="E25">A22</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E19:3_1_1_1_1"/><b>then </b><i><font color="Green" title="E26">A23</font></i>: 
<font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E7:3_1_1_1_1"><i><font color="Green" title="E18">A15</font></i></a>, <a class="txt" href="weddwitt.html#E9:3_1_1_1_1"><i><font color="Green" title="E20">A17</font></i></a>, <a class="txt" href="weddwitt.html#E10:3_1_1_1_1"><i><font color="Green" title="E21">A18</font></i></a>, <a class="txt" href="weddwitt.html#E11:3_1_1_1_1"><i><font color="Green" title="E22">A19</font></i></a>, <a class="txt" href="weddwitt.html#E12:3_1_1_1_1"><i><font color="Green" title="E23">A20</font></i></a>, <a class="txt" href="weddwitt.html#E14:3_1_1_1_1"><i><font color="Green" title="E24">A21</font></i></a>, <a class="ref" href="real_2.html#T144" title="REAL_2:th.144">REAL_2:144</a></i>;<br/><a NAME="E20:3_1_1_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E4:3_1_1_1_1"><i><font color="Green" title="E17">A14</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E21:3_1_1_1_1"/><b>then </b>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E22:3_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E23:3_1_1_1_1"/><b>then </b><i><font color="Green" title="E27">A24</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>
 ;<br/><a NAME="E24:3_1_1_1_1"/>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E4:3_1_1_1_1"><i><font color="Green" title="E17">A14</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E25:3_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E26:3_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>
 ;<br/><a NAME="E27:3_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="weddwitt.html#E19:3_1_1_1_1"><i><font color="Green" title="E26">A23</font></i></a>, <a class="txt" href="weddwitt.html#E23:3_1_1_1_1"><i><font color="Green" title="E27">A24</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E28:3_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="weddwitt.html#E9:3"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="weddwitt.html#E1:3_1_1_1"><i><font color="Green" title="E13">A10</font></i></a>, <a class="txt" href="weddwitt.html#E2:3_1_1_1_1"><i><font color="Green" title="E16">A13</font></i></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/></div><b>end;</b></div><a NAME="E3:3_1_1_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E5:3"><i><font color="Green" title="E8">A5</font></i></a>, <a class="ref" href="prepower.html#T13" title="PREPOWER:th.13">PREPOWER:13</a></i>;<br/><a NAME="E4:3_1_1_1"/><b>then </b>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="weddwitt.html#E2:3_1_1_1"><i><font color="Green" title="E14">A11</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E6:3_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="weddwitt.html#E2:3"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="weddwitt.html#T1" target="_self" title="WEDDWITT:th.1">Th1</a></i>;<br/><b>hence </b><a NAME="E7:3_1_1_1"/>
<font color="Maroon" title="c2">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E6:3"><i><font color="Green" title="E9">A6</font></i></a>, <a class="ref" href="nat_d.html#T3" title="NAT_D:th.3">NAT_D:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_2"><b>suppose </b></a><a NAME="E1:3_1_1_2"/><i><font color="Green" title="E13">A25</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:3_1_1_2"/><b>then </b><i><font color="Green" title="E14">A26</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:3_1_1_2"/><i><font color="Green" title="E15">A27</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="weddwitt.html#E1:3_1_1_2"><i><font color="Green" title="E13">A25</font></i></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/><a NAME="E5:3_1_1_2"/><i><font color="Green" title="E16">A28</font></i>: 
<font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E6:3_1_1_2"/><i><font color="Green" title="E17">A29</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E4:3_1_1_2"><i><font color="Green" title="E15">A27</font></i></a></i>;<br/><a NAME="E7:3_1_1_2"/>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E2:3_1_1_2"><i><font color="Green" title="E14">A26</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E8:3_1_1_2"/><b>then </b><i><font color="Green" title="E18">A30</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E6:3_1_1_2"><i><font color="Green" title="E17">A29</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><b>set </b><font color="Maroon" title="c9">qNaj</font> = <font color="Maroon" title="c1">q</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">j</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon" title="c10">qNak</font> = <font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon" title="c11">qNak1</font> = <font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>;<br/><a NAME="E9:3_1_1_2"/>
<font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E4:3_1_1_2"><i><font color="Green" title="E15">A27</font></i></a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E10:3_1_1_2"/><b>then </b><i><font color="Green" title="E19">A31</font></i>: 
not <span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">j</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="weddwitt.html#E5:3_1_1_2"><i><font color="Green" title="E16">A28</font></i></a>, <a class="txt" href="weddwitt.html#E10:3"><i><font color="Green" title="E12">A9</font></i></a></i>;<br/><a NAME="E11:3_1_1_2"/><i><font color="Green" title="E20">A32</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="weddwitt.html#E9:3"><i><font color="Green" title="E11">A8</font></i></a>, <a class="ref" href="prepower.html#T109" title="PREPOWER:th.109">PREPOWER:109</a></i>;<br/><a NAME="E12:3_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="int_2.html#T14" title="INT_2:th.14">INT_2:14</a></i>;<br/><a NAME="E13:3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E11:3_1_1_2"><i><font color="Green" title="E20">A32</font></i></a>, <a class="ref" href="wsierp_1.html#T9" title="WSIERP_1:th.9">WSIERP_1:9</a></i>;<br/><a NAME="E14:3_1_1_2"/><b>then </b><i><font color="Green" title="E21">A33</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 ;<br/><a NAME="E15:3_1_1_2"/><i><font color="Green" title="E22">A34</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E5:3"><i><font color="Green" title="E8">A5</font></i></a>, <a class="ref" href="prepower.html#T13" title="PREPOWER:th.13">PREPOWER:13</a></i>;<br/><a NAME="E16:3_1_1_2"/><i><font color="Green" title="E23">A35</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E5:3"><i><font color="Green" title="E8">A5</font></i></a>, <a class="ref" href="prepower.html#T13" title="PREPOWER:th.13">PREPOWER:13</a></i>;<br/><a NAME="E17:3_1_1_2"/>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E15:3_1_1_2"><i><font color="Green" title="E22">A34</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E18:3_1_1_2"/><b>then </b><i><font color="Green" title="E24">A36</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E19:3_1_1_2"/><i><font color="Green" title="E25">A37</font></i>: 
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E16:3_1_1_2"><i><font color="Green" title="E23">A35</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E20:3_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span>
 ;<br/><a NAME="E21:3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E2:3"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="weddwitt.html#E2:3_1_1_2"><i><font color="Green" title="E14">A26</font></i></a>, <a class="ref" href="weddwitt.html#T2" target="_self" title="WEDDWITT:th.2">Th2</a></i>;<br/><a NAME="E22:3_1_1_2"/><b>then </b><i><font color="Green" title="E26">A38</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E18:3_1_1_2"><i><font color="Green" title="E24">A36</font></i></a>, <a class="txt" href="weddwitt.html#E19:3_1_1_2"><i><font color="Green" title="E25">A37</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E23:3_1_1_2"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E5:3"><i><font color="Green" title="E8">A5</font></i></a>, <a class="ref" href="prepower.html#T13" title="PREPOWER:th.13">PREPOWER:13</a></i>;<br/><a NAME="E24:3_1_1_2"/><b>then </b>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E25:3_1_1_2"/><b>then </b><i><font color="Green" title="E27">A39</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="weddwitt.html#E22:3_1_1_2"><i><font color="Green" title="E26">A38</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/><a NAME="E26:3_1_1_2"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E5:3"><i><font color="Green" title="E8">A5</font></i></a>, <a class="ref" href="prepower.html#T13" title="PREPOWER:th.13">PREPOWER:13</a></i>;<br/><a NAME="E27:3_1_1_2"/><b>then </b>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E28:3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><a NAME="E29:3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1,<font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="pepin.html#T1" title="PEPIN:th.1">PEPIN:1</a></i>;<br/><a NAME="E30:3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1,<font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="euler_2.html#T1" title="EULER_2:th.1">EULER_2:1</a></i>;<br/><a NAME="E31:3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">q</font> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="weddwitt.html#E14:3_1_1_2"><i><font color="Green" title="E21">A33</font></i></a>, <a class="txt" href="weddwitt.html#E25:3_1_1_2"><i><font color="Green" title="E27">A39</font></i></a>, <a class="ref" href="int_2.html#T40" title="INT_2:th.40">INT_2:40</a></i>;<br/><b>hence </b><a NAME="E32:3_1_1_2"/>
<font color="Maroon" title="c2">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="weddwitt.html#E8:3_1_1_2"><i><font color="Green" title="E18">A30</font></i></a>, <a class="txt" href="weddwitt.html#E10:3_1_1_2"><i><font color="Green" title="E19">A31</font></i></a>, <a class="ref" href="prepower.html#T109" title="PREPOWER:th.109">PREPOWER:109</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E12:3"/>
<font color="Maroon" title="c2">a</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">b</font>
 ;<br/>


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