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<b>let </b><font color="Maroon" title="c1">a</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">l</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">l</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  holds <br/><font color="Olive" title="b1">k</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">a</font>,<font color="Olive" title="b2">l</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> </span><br/><b>let </b><font color="Maroon" title="c2">k</font>, <font color="Maroon" title="c3">l</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">l</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  implies <font color="Maroon" title="c2">k</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">l</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  )</span><br/>





<b>assume </b><a NAME="E1:14"/><i><font color="Green" title="E14">A1</font></i>: 
<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">l</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">k</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">l</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> </span><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="c2">k</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> $1,<font color="Maroon" title="c3">l</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> ;<br/>
<a NAME="E2:14"/>
<font color="Maroon" title="c2">k</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>;<br/>
<a NAME="E3:14"/><b>then </b><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="ref" href="wsierp_1.html#T14" target="_self" title="WSIERP_1:th.14">Th14</a></i>;<br/>
<a NAME="E4:14"/><i><font color="Green" title="E16">A3</font></i>: 
for <font color="Olive" title="b1">a</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">a</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">a</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">a</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:14_1"/>
<font color="Maroon" title="c2">k</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">l</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">a</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E2:14_1"/><b>then </b>
<font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">a</font></span>)</span>,<font color="Maroon" title="c3">l</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="#E1:14"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="int_2.html#T41" title="INT_2:th.41">INT_2:41</a></i>;<br/>
<b>hence </b><a NAME="E3:14_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">a</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:14"/>
for <font color="Olive" title="b1">a</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">a</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="#E3:14"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="#E4:14"><i><font color="Green" title="E16">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:14"/>
<font color="Maroon" title="c2">k</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">l</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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