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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:23"/><i><font color="Green" title="E16">A1</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Maroon" title="c2">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">m</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a>  )
 ;<br/>

<a NAME="E2:23"/><i><font color="Green" title="E17">A2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">m</font>
 
<b>by </b><i><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> $1,<font color="Maroon" title="c2">m</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> ;<br/>
<a NAME="E3:23"/>
<font color="Maroon" title="c2">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="newton.html#T64" title="NEWTON:th.64">NEWTON:64</a></i>;<br/>
<a NAME="E4:23"/><b>then </b><i><font color="Green" title="E18">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ]
 
<b>by </b><i><a class="txt" href="euler_2.html#E2:23"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/>
<a NAME="E5:23"/><i><font color="Green" title="E19">A4</font></i>: 
for <font color="Olive" title="b1">b</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">b</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">b</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">b</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:23_1"/><i><font color="Green" title="E20">A5</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">m</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 ;<br/>

<i><font color="Green" title="E21">A6</font></i>: <font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">a</font>
<b>by </b><i><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></i>
;<br/>
<a NAME="E3:23_1"/>
<font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<b>by </b><i><a class="txt" href="euler_2.html#E1:23"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="prepower.html#T12" title="PREPOWER:th.12">PREPOWER:12</a></i>;<br/>
<b>hence </b><a NAME="E4:23_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">b</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="euler_2.html#E1:23"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="euler_2.html#E1:23_1"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="euler_2.html#E2:23_1"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="euler_1.html#T15" title="EULER_1:th.15">EULER_1:15</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:23"/>
for <font color="Olive" title="b1">b</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">b</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="euler_2.html#E4:23"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="euler_2.html#E5:23"><i><font color="Green" title="E19">A4</font></i></a>);<br/></i>
<b>hence </b><a NAME="E7:23"/>
for <font color="Olive" title="b1">b</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c1">a</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b1">b</font>,<font color="Maroon" title="c2">m</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 ;<br/>


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