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



<b>given </b><font color="Maroon">q</font> being   <a href="rat_1.html#NM1">Rational</a><b> such that </b><a NAME="E1:61"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="newton.html#K2">|^</a> <font color="Maroon">b</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>now </b></a><div class="add"><div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:61_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="wsierp_1.html#E7"><i><font color="Green">Lemma44</font></i></a>, <a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/><a NAME="E3:61_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1 <a href="wsierp_1.html#K2" target="_self">|^</a> 0
 <b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/><b>hence </b><a NAME="E4:61_1_1"/>
 ex <font color="Olive">k</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> <a href="newton.html#K2">|^</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E9"><i><font color="Green">Lemma46</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:61_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><b>set </b><font color="Maroon">k</font> =  <a href="rat_1.html#K2">numerator</a> <font color="Maroon">q</font>;<br/><b>set </b><font color="Maroon">d</font> =  <a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font>;<br/><a NAME="E2:61_1_2"/><i><font color="Green">E50</font></i>: 
 <a href="rat_1.html#K2">numerator</a> <font color="Maroon">q</font>, <a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font> <a href="int_2.html#R1">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="wsierp_1.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E3:61_1_2"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2">numerator</a> <font color="Maroon">q</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="rat_1.html#T37">RAT_1:37</a></i>;<br/><a NAME="E4:61_1_2"/>
 <a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="rat_1.html#D3">RAT_1:def 3</a></i>;<br/><a NAME="E5:61_1_2"/><b>then </b><i><font color="Green">E105</font></i>: 
<span class="p1">(<span class="default"><a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="card_4.html#T51">CARD_4:51</a></i>;<br/><a NAME="E6:61_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K2">numerator</a> <font color="Maroon">q</font></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E7"><i><font color="Green">Lemma44</font></i></a>, , <a class="ref" href="prepower.html#T15">PREPOWER:15</a></i>;<br/><a NAME="E7:61_1_2"/><b>then </b><i><font color="Green">E106</font></i>: 
<font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font></span>)</span> <a href="wsierp_1.html#K2" target="_self">|^</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2">numerator</a> <font color="Maroon">q</font></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E1:61_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><a NAME="E8:61_1_2"/>
<span class="p1">(<span class="default"><a href="rat_1.html#K2">numerator</a> <font color="Maroon">q</font></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">b</font>, <a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font> <a href="int_2.html#R1">are_relative_prime</a> 
 <b>by </b><i><a class="txt" href="wsierp_1.html#E12"><i><font color="Green">Lemma47</font></i></a>, </i>;<br/><a NAME="E9:61_1_2"/><b>then </b><i><font color="Green">E107</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K2">numerator</a> <font color="Maroon">q</font></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">b</font></span>)</span> <a href="int_2.html#K3">gcd</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="int_2.html#D4">INT_2:def 4</a></i>;<br/><b>consider </b><font color="Maroon">e</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E11:61_1_2"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">e</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E9"><i><font color="Green">Lemma46</font></i></a>, <a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><span class="p1">(<span class="default"><a href="rat_1.html#K2">numerator</a> <font color="Maroon">q</font></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">b</font> = 
<font color="Maroon">a</font> <a href="real_1.html#K4">*</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">denominator</a> <font color="Maroon">q</font></span>)</span> <a href="newton.html#K3">|^</a> <font color="Maroon">e</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="wsierp_1.html#E2:61_1_2"><i><font color="Green">E50</font></i></a>, , <a class="ref" href="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font></span>)</span> <a href="newton.html#K3">|^</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font></span>)</span>

;<br/><a NAME="E13:61_1_2"/><b>then </b>
 <a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font> <a href="int_1.html#R2">divides</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2">numerator</a> <font color="Maroon">q</font></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/><a NAME="E14:61_1_2"/><b>then </b>
 <a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font> <a href="int_1.html#R2">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rat_1.html#K2">numerator</a> <font color="Maroon">q</font></span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">b</font></span>)</span> <a href="int_2.html#K3">gcd</a> <span class="p1">(<span class="default"><a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="int_2.html#T33">INT_2:33</a></i>;<br/><a NAME="E15:61_1_2"/><b>then </b>
(  <a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> 1 or  <a href="rat_1.html#K1">denominator</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 <b>by </b><i><a class="txt" href="wsierp_1.html#E13"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="int_2.html#T17">INT_2:17</a></i>;<br/><b>hence </b><a NAME="E16:61_1_2"/>
 ex <font color="Olive">k</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> <a href="newton.html#K2">|^</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E7"><i><font color="Green">Lemma44</font></i></a>, , <a class="ref" href="int_2.html#T9">INT_2:9</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:61_1"/>
 ex <font color="Olive">k</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> <a href="newton.html#K2">|^</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="wsierp_1.html#E8"><i><font color="Green">Lemma45</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:61"/>
 ex <font color="Olive">k</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> <a href="newton.html#K2">|^</a> <font color="Maroon">b</font>
 ;<br/>


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