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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:23"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  )
 ;<br/>

<a NAME="E2:23"/><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> 0</span>)</span> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="int_2.html#R2">are_relative_prime</a> ;<br/>
<a NAME="E3:23"/>
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K8">hcf</a> 1 <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="newton.html#T64">NEWTON:64</a></i>;<br/>
<a NAME="E4:23"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="txt" href="euler_2.html#E2:23"><i><font color="Green">E20</font></i></a>, <a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<a NAME="E5:23"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</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">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:23_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 ;<br/>

<i><font color="Green">E24</font></i>: <font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="newton.html#T10">NEWTON:10</a></i>
;<br/>
<a NAME="E3:23_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="euler_2.html#E1:23"><i><font color="Green">E19</font></i></a>, <a class="ref" href="prepower.html#T12">PREPOWER:12</a></i>;<br/>
<b>hence </b><a NAME="E4:23_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="euler_2.html#E1:23"><i><font color="Green">E19</font></i></a>, <a class="txt" href="euler_2.html#E1:23_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="euler_2.html#E2:23_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="euler_1.html#T15">EULER_1:15</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:23"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="euler_2.html#E4:23"><i><font color="Green">E21</font></i></a>, <a class="txt" href="euler_2.html#E5:23"><i><font color="Green">E22</font></i></a>);<br/></i>
<b>hence </b><a NAME="E7:23"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 ;<br/>


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