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<b>let </b><font color="Maroon">q</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">a</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">b</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:3"/><i><font color="Green">E35</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E2:3"/><i><font color="Green">E36</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q</font>
 <b>and </b><br/><a NAME="E3:3"/><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">b</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 ;<br/>

<a NAME="E4:3"/><i><font color="Green">E38</font></i>: 
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:3"/><b>then </b><i><font color="Green">E46</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<b>set </b><font color="Maroon">r</font> = <font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font>;<br/>
<b>set </b><font color="Maroon">qNa</font> = <font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font>;<br/>
<b>set </b><font color="Maroon">qNr</font> = <font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1;<br/>
<a NAME="E6:3"/><i><font color="Green">E50</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="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T47">NAT_1:47</a></i>;<br/>
<a NAME="E7:3"/><b>then </b><i><font color="Green">E51</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:3"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 <b>and </b><br/><a NAME="E10:3"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</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">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k</font> <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="3_1_1_1"><b>suppose </b></a><a NAME="E1:3_1_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><div><i><font color="Green">E56</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">E57</font></i>: 
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
 ;<br/><a NAME="E2:3_1_1_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><a NAME="E3:3_1_1_1_1"/>
<font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="nat_2.html#T10">NAT_2:10</a></i>;<br/><a NAME="E4:3_1_1_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><a NAME="E5:3_1_1_1_1"/>
<font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/><b>then consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:3_1_1_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E8:3_1_1_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/><a NAME="E9:3_1_1_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">q</font> <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="prepower.html#K7">#Z</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="prepower.html#T46">PREPOWER:46</a></i>;<br/><a NAME="E10:3_1_1_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">q</font> <a href="prepower.html#K7">#Z</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="prepower.html#K7">#Z</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="prepower.html#K7">#Z</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="prepower.html#T54">PREPOWER:54</a></i>;<br/><a NAME="E11:3_1_1_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">q</font> <a href="prepower.html#K7">#Z</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="prepower.html#T46">PREPOWER:46</a></i>;<br/><a NAME="E12:3_1_1_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">q</font> <a href="prepower.html#K7">#Z</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="newton.html#K3">|^</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="prepower.html#T46">PREPOWER:46</a></i>;<br/><a NAME="E13:3_1_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><a NAME="E14:3_1_1_1_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">q</font> <a href="prepower.html#K7">#Z</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="prepower.html#T49">PREPOWER:49</a></i>;<br/><a NAME="E15:3_1_1_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">q</font> <a href="newton.html#K3">|^</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i>, <a class="ref" href="prepower.html#T19">PREPOWER:19</a></i>;<br/><a NAME="E16:3_1_1_1_1"/>
<font color="Maroon">m</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="E17:3_1_1_1_1"/><b>then </b>
<font color="Maroon">q</font> <a href="newton.html#K3">|^</a> <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>by </b><i>, , </i>;<br/><a NAME="E18:3_1_1_1_1"/><b>then </b>
<font color="Maroon">q</font> <a href="newton.html#K3">|^</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E19:3_1_1_1_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font>
 <b>by </b><i>, , , , , , <a class="ref" href="real_2.html#T144">REAL_2:144</a></i>;<br/><a NAME="E20:3_1_1_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E21:3_1_1_1_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E22:3_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E23:3_1_1_1_1"/><b>then </b><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 ;<br/><a NAME="E24:3_1_1_1_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E25:3_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T50">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">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</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">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E28:3_1_1_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/></div><b>end;</b></div><a NAME="E3:3_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="prepower.html#T13">PREPOWER:13</a></i>;<br/><a NAME="E4:3_1_1_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</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">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E6:3_1_1_1"/><b>then </b>
<font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E7:3_1_1_1"/>
<font color="Maroon">a</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T49">NAT_1:49</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">E71</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:3_1_1_2"/><b>then </b><i><font color="Green">E72</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/><b>consider </b><font color="Maroon">j</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:3_1_1_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</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/><a NAME="E5:3_1_1_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">j</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="E6:3_1_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 <b>by </b><i/>;<br/><a NAME="E7:3_1_1_2"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:3_1_1_2"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><b>set </b><font color="Maroon">qNaj</font> = <font color="Maroon">q</font> <a href="newton.html#K3">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">qNak</font> = <font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">qNak1</font> = <font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span></span>)</span>;<br/><a NAME="E9:3_1_1_2"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E10:3_1_1_2"/><b>then </b><i><font color="Green">E81</font></i>: 
not <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="newton.html#K3">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i>, </i>;<br/><a NAME="E11:3_1_1_2"/><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="int_1.html#R2">divides</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i>, <a class="ref" href="prepower.html#T109">PREPOWER:109</a></i>;<br/><a NAME="E12:3_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="int_1.html#R2">divides</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="int_2.html#T14">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">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="int_1.html#R2">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="wsierp_1.html#T9">WSIERP_1:9</a></i>;<br/><a NAME="E14:3_1_1_2"/><b>then </b><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="int_1.html#R2">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 ;<br/><a NAME="E15:3_1_1_2"/><i><font color="Green">E84</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="prepower.html#T13">PREPOWER:13</a></i>;<br/><a NAME="E16:3_1_1_2"/><i><font color="Green">E85</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="prepower.html#T13">PREPOWER:13</a></i>;<br/><a NAME="E17:3_1_1_2"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E18:3_1_1_2"/><b>then </b><i><font color="Green">E86</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E19:3_1_1_2"/><i><font color="Green">E87</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</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">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">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">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span></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"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> 1</span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E22:3_1_1_2"/><b>then </b><i><font color="Green">E88</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E23:3_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="prepower.html#T13">PREPOWER:13</a></i>;<br/><a NAME="E24:3_1_1_2"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E25:3_1_1_2"/><b>then </b><i><font color="Green">E89</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E26:3_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="prepower.html#T13">PREPOWER:13</a></i>;<br/><a NAME="E27:3_1_1_2"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</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">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E29:3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 1,<font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="pepin.html#T1">PEPIN:1</a></i>;<br/><a NAME="E30:3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 1,<font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font> <a href="int_2.html#R1">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="euler_2.html#T1">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">q</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="int_1.html#R2">divides</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i>, , <a class="ref" href="int_2.html#T40">INT_2:40</a></i>;<br/><b>hence </b><a NAME="E32:3_1_1_2"/>
<font color="Maroon">a</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font>
 <b>by </b><i>, , <a class="ref" href="prepower.html#T109">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">a</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font>
 ;<br/>


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