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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/>





<b>assume </b><a NAME="E1:19"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="int_2.html#R1">are_relative_prime</a> 
 ;<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>2</sub></font> <a href="newton.html#K2">|^</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="int_2.html#R1">are_relative_prime</a> ;<br/>
<a NAME="E2:19"/>
<font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K2">|^</a> 0 <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/>
<a NAME="E3:19"/><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="ref" href="wsierp_1.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E4:19"/><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="19_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:19_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="int_2.html#R1">are_relative_prime</a> 
 ;<br/>

<a NAME="E2:19_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="int_2.html#R1">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="wsierp_1.html#E1:19"><i><font color="Green">E20</font></i></a>, <a class="ref" href="int_2.html#T41">INT_2:41</a></i>;<br/>
<b>hence </b><a NAME="E3:19_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:19"/>
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="wsierp_1.html#E3:19"><i><font color="Green">E21</font></i></a>, <a class="txt" href="wsierp_1.html#E4:19"><i><font color="Green">E22</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:19"/>
<font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="int_2.html#R1">are_relative_prime</a> 
 ;<br/>


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