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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">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>given </b><font color="Maroon" title="c3">q</font> being   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a><b> such that </b><a NAME="E1:60"/><i><font color="Green" title="E44">A1</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">b</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>now </b></a><div class="add"><div><i><font color="Green" title="E45">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:60_1_1"/><i><font color="Green" title="E46">A3</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><a NAME="E2:60_1_1"/><b>then </b>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:60"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>;<br/><a NAME="E3:60_1_1"/><b>then </b>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> 0
 <b>by </b><i><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>;<br/><b>hence </b><a NAME="E4:60_1_1"/>
 ex <font color="Olive" title="b1">k</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:60_1_1"><i><font color="Green" title="E46">A3</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:60_1_2"/><i><font color="Green" title="E46">A4</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><b>set </b><font color="Maroon" title="c4">k</font> =  <a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font>;<br/><b>set </b><font color="Maroon" title="c5">d</font> =  <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font>;<br/><a NAME="E2:60_1_2"/><i><font color="Green" title="E47">A5</font></i>: 
 <a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font>, <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="wsierp_1.html#T29" target="_self" title="WSIERP_1:th.29">Th29</a></i>;<br/><a NAME="E3:60_1_2"/><i><font color="Green" title="E48">A6</font></i>: 
<font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="rat_1.html#T37" title="RAT_1:th.37">RAT_1:37</a></i>;<br/><a NAME="E4:60_1_2"/>
 <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="rat_1.html#D3" title="RAT_1:def.3">RAT_1:def 3</a></i>;<br/><a NAME="E5:60_1_2"/><b>then </b><i><font color="Green" title="E49">A7</font></i>: 
<span class="p1">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="card_4.html#T51" title="CARD_4:th.51">CARD_4:51</a></i>;<br/><a NAME="E6:60_1_2"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:60"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="wsierp_1.html#E3:60_1_2"><i><font color="Green" title="E48">A6</font></i></a>, <a class="ref" href="prepower.html#T15" title="PREPOWER:th.15">PREPOWER:15</a></i>;<br/><a NAME="E7:60_1_2"/><b>then </b><i><font color="Green" title="E50">A8</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">|^</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E5:60_1_2"><i><font color="Green" title="E49">A7</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>;<br/><a NAME="E8:60_1_2"/>
<span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">b</font>, <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 <b>by </b><i><a class="txt" href="wsierp_1.html#E2:60_1_2"><i><font color="Green" title="E47">A5</font></i></a>, <a class="ref" href="wsierp_1.html#T15" target="_self" title="WSIERP_1:th.15">Th15</a></i>;<br/><a NAME="E9:60_1_2"/><b>then </b><i><font color="Green" title="E51">A9</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c6">e</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E11:60_1_2"/><i><font color="Green" title="E52">A10</font></i>: 
<font color="Maroon" title="c6">e</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:60_1_2"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/><span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">b</font> = 
<font color="Maroon" title="c1">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c6">e</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E7:60_1_2"><i><font color="Green" title="E50">A8</font></i></a>, <a class="txt" href="wsierp_1.html#E11:60_1_2"><i><font color="Green" title="E52">A10</font></i></a>, <a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c6">e</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span>

;<br/><a NAME="E13:60_1_2"/><b>then </b>
 <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></i>;<br/><a NAME="E14:60_1_2"/><b>then </b>
 <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font> <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"><a href="rat_1.html#K2" title="RAT_1:func.2">numerator</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="int_2.html#T33" title="INT_2:th.33">INT_2:33</a></i>;<br/><a NAME="E15:60_1_2"/><b>then </b>
(  <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="rat_1.html#K1" title="RAT_1:func.1">denominator</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1 )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E9:60_1_2"><i><font color="Green" title="E51">A9</font></i></a>, <a class="ref" href="int_2.html#T17" title="INT_2:th.17">INT_2:17</a></i>;<br/><b>hence </b><a NAME="E16:60_1_2"/>
 ex <font color="Olive" title="b1">k</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:60"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="wsierp_1.html#E3:60_1_2"><i><font color="Green" title="E48">A6</font></i></a>, <a class="ref" href="int_2.html#T9" title="INT_2:th.9">INT_2:9</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:60_1"/>
 ex <font color="Olive" title="b1">k</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">b</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:60_1"><i><font color="Green" title="E45">A2</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:60"/>
 ex <font color="Olive" title="b1">k</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">b</font>
 ;<br/>


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