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<span class="kw">let </span><font color="Maroon" title="c1">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>  holds <br/>( <font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  or <font color="Olive" title="b1">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">i</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font> be   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  or <font color="Maroon" title="c2">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">i</font> )</span><br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:16_1"/>
( <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1"><span class="kw">suppose </span></a><a NAME="E1:16_1_1"/>
<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  or <font color="Maroon" title="c2">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">i</font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">i</font> = <font color="Maroon" title="c1">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></span>;<br/><a NAME="E3:16_1_1"/>
( <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c2">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  or <font color="Maroon" title="c3">i</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pepin.html#T2" title="PEPIN:th.2">PEPIN:2</a></span>;<br/><span class="kw">hence </span><a NAME="E4:16_1_1"/>
( <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  or <font color="Maroon" title="c2">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2"><span class="kw">suppose </span></a><a NAME="E1:16_1_2"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  or <font color="Maroon" title="c2">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">i</font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">m</font> =  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></span>;<br/><a NAME="E3:16_1_2"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
( <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c2">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  or <font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pepin.html#T2" title="PEPIN:th.2">PEPIN:2</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:16_1_2_1"/>
( <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c2">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  or <font color="Maroon" title="c2">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">m</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_2"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:16_1_2_1_1"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c2">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  or <font color="Maroon" title="c2">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">i</font> )</span><br/><div class="add"><a NAME="E2:16_1_2_1_1"/>
<font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c1">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_2"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c1">i</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">p</font> = 
<font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c2">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T51" title="INT_2:th.51">INT_2:51</a></span>
<br/>.= 
<font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_2_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:16_1_2_1_1"/>
( <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  or <font color="Maroon" title="c2">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:16_1_2_1_2"/>
<font color="Maroon" title="c2">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  or <font color="Maroon" title="c2">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">i</font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">t</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:16_1_2_1_2"/><span class="lab"><font color="Green" title="E16">A4</font></span>: 
<font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">t</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></span>;<br/><a NAME="E4:16_1_2_1_2"/>
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c4">t</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_2_1_2"><span class="lab"><font color="Green" title="E16">A4</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:16_1_2_1_2"/>
( <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  or <font color="Maroon" title="c2">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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