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<b>let </b><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">q</font>, <font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">k1</font>, <font color="Maroon" title="c5">k2</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" title="c1">p</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> &amp; <font color="Maroon" title="c2">q</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> &amp; <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">q</font> &amp; <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">q</font> &amp; <font color="Maroon" title="c4">k1</font>, <a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c3">n</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">k1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">k2</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font> holds <br/> <a href="pepin.html#K3" title="PEPIN:func.3">Crypto</a> <span class="p1">(<span class="default"><a href="pepin.html#K3" title="PEPIN:func.3">Crypto</a> <font color="Olive" title="b1">m</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c4">k1</font></span>)</span>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c5">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> )</span><br/>









<b>assume </b><a NAME="E1:55"/><i><font color="Green" title="E41">A1</font></i>: 
( <font color="Maroon" title="c1">p</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> &amp; <font color="Maroon" title="c2">q</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> &amp; <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">q</font> &amp; <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">q</font> &amp; <font color="Maroon" title="c4">k1</font>, <a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c3">n</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">k1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">k2</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ; <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">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font> holds <br/> <a href="pepin.html#K3" title="PEPIN:func.3">Crypto</a> <span class="p1">(<span class="default"><a href="pepin.html#K3" title="PEPIN:func.3">Crypto</a> <font color="Olive" title="b1">m</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c4">k1</font></span>)</span>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c5">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font></span><br/>

<b>let </b><font color="Maroon" title="c6">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font> implies  <a href="pepin.html#K3" title="PEPIN:func.3">Crypto</a> <span class="p1">(<span class="default"><a href="pepin.html#K3" title="PEPIN:func.3">Crypto</a> <font color="Maroon" title="c6">m</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c4">k1</font></span>)</span>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c5">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font> )</span><br/>

<b>assume </b><a NAME="E2:55"/><i><font color="Green" title="E42">A2</font></i>: 
<font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="pepin.html#K3" title="PEPIN:func.3">Crypto</a> <span class="p1">(<span class="default"><a href="pepin.html#K3" title="PEPIN:func.3">Crypto</a> <font color="Maroon" title="c6">m</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c4">k1</font></span>)</span>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c5">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font></span><br/>

<a NAME="E3:55"/><i><font color="Green" title="E43">A3</font></i>: 
( <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="#E1:55"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">p1</font> = <font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1, <font color="Maroon" title="c8">q1</font> = <font color="Maroon" title="c2">q</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="pepin.html#E3"><i><font color="Green" title="E3">Lm3</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<a NAME="E5:55"/><i><font color="Green" title="E44">A4</font></i>: 
( <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c2">q</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 
<b>by </b><i><a class="txt" href="#E1:55"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>
<a NAME="E6:55"/><i><font color="Green" title="E45">A5</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E1:55"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="#E3:55"><i><font color="Green" title="E43">A3</font></i></a>, <a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></i>;<br/>
<a NAME="E7:55"/><i><font color="Green" title="E46">A6</font></i>: 
( <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">q</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="#E1:55"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>
<a NAME="E8:55"/>
 <a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E6:55"><i><font color="Green" title="E45">A5</font></i></a>, <a class="ref" href="pepin.html#T45" target="_self" title="PEPIN:th.45">Th45</a></i>;<br/>
<a NAME="E9:55"/><b>then </b><i><font color="Green" title="E47">A7</font></i>: 
 <a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<i><font color="Green" title="E48">A8</font></i>:  <a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c3">n</font> = 
<span class="p1">(<span class="default"><a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c2">q</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:55"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="#E5:55"><i><font color="Green" title="E44">A4</font></i></a>, <a class="ref" href="euler_1.html#T22" title="EULER_1:th.22">EULER_1:22</a>, <a class="ref" href="int_2.html#T47" title="INT_2:th.47">INT_2:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c2">q</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:55"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="euler_1.html#T21" title="EULER_1:th.21">EULER_1:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="#E1:55"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="euler_1.html#T21" title="EULER_1:th.21">EULER_1:21</a></i>
;<br/>
<a NAME="E11:55"/>
(  not <font color="Maroon" title="c7">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  not <font color="Maroon" title="c8">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="#E1:55"><i><font color="Green" title="E41">A1</font></i></a></i>;<br/>
<a NAME="E12:55"/><b>then </b><i><font color="Green" title="E49">A9</font></i>: 
<font color="Maroon" title="c7">p1</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T15" title="NAT_1:th.15">NAT_1:15</a></i>;<br/>
<a NAME="E13:55"/><i><font color="Green" title="E50">A10</font></i>: 
<font color="Maroon" title="c4">k1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:55_2"/>
<font color="Maroon" title="c4">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:55_2"/><b>then </b>
<font color="Maroon" title="c4">k1</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <span class="p1">(<span class="default"><a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="ref" href="newton.html#T65" title="NEWTON:th.65">NEWTON:65</a></i>;<br/>
<b>hence </b><a NAME="E3:55_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:55"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="#E10:55"><i><font color="Green" title="E48">A8</font></i></a>, <a class="txt" href="#E12:55"><i><font color="Green" title="E49">A9</font></i></a>, <a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</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="E14:55"/><i><font color="Green" title="E51">A11</font></i>: 
<font color="Maroon" title="c5">k2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E1:55"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="nat_d.html#T26" title="NAT_D:th.26">NAT_D:26</a></i>;<br/>
<a NAME="E15:55"/><i><font color="Green" title="E52">A12</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="#E1:55"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="#E5:55"><i><font color="Green" title="E44">A4</font></i></a>, <a class="ref" href="xreal_1.html#T163" title="XREAL_1:th.163">XREAL_1:163</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_3_1"/>
( <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or ( <font color="Maroon" title="c6">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c6">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_3_1_1"><b>suppose </b></a><a NAME="E1:55_3_1_1"/><i><font color="Green" title="E53">A13</font></i>: 
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="pepin.html#K3" title="PEPIN:func.3">Crypto</a> <span class="p1">(<span class="default"><a href="pepin.html#K3" title="PEPIN:func.3">Crypto</a> <font color="Maroon" title="c6">m</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c4">k1</font></span>)</span>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c5">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font></span><br/><div class="add"><a NAME="E2:55_3_1_1"/><b>then </b>
<font color="Maroon" title="c6">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E13:55"><i><font color="Green" title="E50">A10</font></i></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="newton.html#T16" title="NEWTON:th.16">NEWTON:16</a></i>;<br/><a NAME="E3:55_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">k1</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="nat_d.html#T26" title="NAT_D:th.26">NAT_D:26</a></i>;<br/><a NAME="E4:55_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pepin.html#K3" title="PEPIN:func.3">Crypto</a> <font color="Maroon" title="c6">m</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c4">k1</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E14:55"><i><font color="Green" title="E51">A11</font></i></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="newton.html#T16" title="NEWTON:th.16">NEWTON:16</a></i>;<br/><b>hence </b><a NAME="E5:55_3_1_1"/>
 <a href="pepin.html#K3" title="PEPIN:func.3">Crypto</a> <span class="p1">(<span class="default"><a href="pepin.html#K3" title="PEPIN:func.3">Crypto</a> <font color="Maroon" title="c6">m</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c4">k1</font></span>)</span>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c5">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font>
 <b>by </b><i><a class="txt" href="#E1:55_3_1_1"><i><font color="Green" title="E53">A13</font></i></a>, <a class="ref" href="nat_d.html#T26" title="NAT_D:th.26">NAT_D:26</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_3_1_2"><b>suppose </b></a><a NAME="E1:55_3_1_2"/><i><font color="Green" title="E53">A14</font></i>: 
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="pepin.html#K3" title="PEPIN:func.3">Crypto</a> <span class="p1">(<span class="default"><a href="pepin.html#K3" title="PEPIN:func.3">Crypto</a> <font color="Maroon" title="c6">m</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c4">k1</font></span>)</span>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c5">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font></span><br/><div class="add"><a NAME="E2:55_3_1_2"/><b>then </b>
<font color="Maroon" title="c6">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="newton.html#T15" title="NEWTON:th.15">NEWTON:15</a></i>;<br/><a NAME="E3:55_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">k1</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="#E15:55"><i><font color="Green" title="E52">A12</font></i></a>, <a class="ref" href="nat_d.html#T14" title="NAT_D:th.14">NAT_D:14</a></i>;<br/><a NAME="E4:55_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="pepin.html#K3" title="PEPIN:func.3">Crypto</a> <font color="Maroon" title="c6">m</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c4">k1</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="newton.html#T15" title="NEWTON:th.15">NEWTON:15</a></i>;<br/><b>hence </b><a NAME="E5:55_3_1_2"/>
 <a href="pepin.html#K3" title="PEPIN:func.3">Crypto</a> <span class="p1">(<span class="default"><a href="pepin.html#K3" title="PEPIN:func.3">Crypto</a> <font color="Maroon" title="c6">m</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c4">k1</font></span>)</span>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c5">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font>
 <b>by </b><i><a class="txt" href="#E15:55"><i><font color="Green" title="E52">A12</font></i></a>, <a class="txt" href="#E1:55_3_1_2"><i><font color="Green" title="E53">A14</font></i></a>, <a class="ref" href="nat_d.html#T14" title="NAT_D:th.14">NAT_D:14</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_3_1_3"><b>suppose </b></a><a NAME="E1:55_3_1_3"/><i><font color="Green" title="E53">A15</font></i>: 
( <font color="Maroon" title="c6">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c6">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="pepin.html#K3" title="PEPIN:func.3">Crypto</a> <span class="p1">(<span class="default"><a href="pepin.html#K3" title="PEPIN:func.3">Crypto</a> <font color="Maroon" title="c6">m</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c4">k1</font></span>)</span>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c5">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font></span><br/><div class="add"><a NAME="E2:55_3_1_3"/><i><font color="Green" title="E54">A16</font></i>: 
for <font color="Olive" title="b1">t</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">t</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">p1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_3_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">t</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">p1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_3_1_3_1_1"><b>now </b></a><div class="add"><a NAME="E1:55_3_1_3_1_1"/>
<font color="Maroon" title="c6">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">p1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="#E1:55_3_1_3"><i><font color="Green" title="E53">A15</font></i></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="prepower.html#T19" title="PREPOWER:th.19">PREPOWER:19</a></i>;<br/><a NAME="E2:55_3_1_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">p1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">q1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">a</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">p1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">q1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E4:55_3_1_3_1_1"/><i><font color="Green" title="E55">A17</font></i>: 
<font color="Maroon" title="c2">q</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_3_1_3_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_3_1_3_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_3_1_3_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_3_1_3_1_1_1_1_1"/>
( <font color="Maroon" title="c6">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c6">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_3_1_3_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:55_3_1_3_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">q</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">a</font></span><br/><div class="add"><a NAME="E2:55_3_1_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E56">A18</font></i>: 
<font color="Maroon" title="c6">m</font>,<font color="Maroon" title="c2">q</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></i>;<br/><a NAME="E3:55_3_1_3_1_1_1_1_1_1"/>
<font color="Maroon" title="c6">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">p1</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E1:55_3_1_3"><i><font color="Green" title="E53">A15</font></i></a>, <a class="ref" href="prepower.html#T12" title="PREPOWER:th.12">PREPOWER:12</a></i>;<br/><a NAME="E4:55_3_1_3_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">p1</font></span>)</span></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">q1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">p1</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">q</font>
 <b>by </b><i><a class="txt" href="#E1:55"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="#E1:55_3_1_3"><i><font color="Green" title="E53">A15</font></i></a>, <a class="txt" href="#E2:55_3_1_3_1_1_1_1_1_1"><i><font color="Green" title="E56">A18</font></i></a>, <a class="ref" href="euler_2.html#T32" title="EULER_2:th.32">EULER_2:32</a>, <a class="ref" href="euler_2.html#T34" title="EULER_2:th.34">EULER_2:34</a></i>;<br/><a NAME="E5:55_3_1_3_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">p1</font></span>)</span></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">p1</font></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">p1</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">q</font>
 <b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>;<br/><a NAME="E6:55_3_1_3_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">p1</font></span>)</span></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="#E5:55"><i><font color="Green" title="E44">A4</font></i></a>, <a class="txt" href="#E1:55_3_1_3"><i><font color="Green" title="E53">A15</font></i></a>, <a class="txt" href="#E2:55_3_1_3_1_1_1_1_1_1"><i><font color="Green" title="E56">A18</font></i></a>, <a class="ref" href="pepin.html#T11" target="_self" title="PEPIN:th.11">Th11</a>, <a class="ref" href="euler_2.html#T32" title="EULER_2:th.32">EULER_2:32</a></i>;<br/><a NAME="E7:55_3_1_3_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">p1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">q1</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="newton.html#T14" title="NEWTON:th.14">NEWTON:14</a></i>;<br/><a NAME="E8:55_3_1_3_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">p1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">p1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">q1</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c2">q</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="#E7:55"><i><font color="Green" title="E46">A6</font></i></a>, <a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></i>;<br/><a NAME="E9:55_3_1_3_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">q</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c10">a</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><b>hence </b><a NAME="E10:55_3_1_3_1_1_1_1_1_1"/>
<font color="Maroon" title="c2">q</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">a</font>
 <b>by </b><i><a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_3_1_3_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:55_3_1_3_1_1_1_1_1_2"/>
<font color="Maroon" title="c6">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">q</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">a</font></span><br/><div class="add"><a NAME="E2:55_3_1_3_1_1_1_1_1_2"/><b>then </b>
not <font color="Maroon" title="c6">m</font>,<font color="Maroon" title="c2">q</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></i>;<br/><a NAME="E3:55_3_1_3_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">q</font>
 <b>by </b><i><a class="txt" href="#E1:55"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="pepin.html#T2" target="_self" title="PEPIN:th.2">Th2</a></i>;<br/><a NAME="E4:55_3_1_3_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">q</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">m</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></i>;<br/><b>hence </b><a NAME="E5:55_3_1_3_1_1_1_1_1_2"/>
<font color="Maroon" title="c2">q</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">a</font>
 <b>by </b><i><a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</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></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:55_3_1_3_1_1_1"/>
<font color="Maroon" title="c2">q</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">a</font>
 ; <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="E5:55_3_1_3_1_1"/>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_3_1_3_1_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_3_1_3_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_3_1_3_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_3_1_3_1_1_2_1_1"/>
( <font color="Maroon" title="c6">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c6">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_3_1_3_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:55_3_1_3_1_1_2_1_1_1"/>
<font color="Maroon" title="c6">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">a</font></span><br/><div class="add"><a NAME="E2:55_3_1_3_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E56">A19</font></i>: 
<font color="Maroon" title="c6">m</font>,<font color="Maroon" title="c1">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></i>;<br/><a NAME="E3:55_3_1_3_1_1_2_1_1_1"/>
<font color="Maroon" title="c6">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E1:55_3_1_3"><i><font color="Green" title="E53">A15</font></i></a>, <a class="ref" href="prepower.html#T12" title="PREPOWER:th.12">PREPOWER:12</a></i>;<br/><a NAME="E4:55_3_1_3_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">q1</font></span>)</span></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">p1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">q1</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="#E1:55"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="#E1:55_3_1_3"><i><font color="Green" title="E53">A15</font></i></a>, <a class="txt" href="#E2:55_3_1_3_1_1_2_1_1_1"><i><font color="Green" title="E56">A19</font></i></a>, <a class="ref" href="euler_2.html#T32" title="EULER_2:th.32">EULER_2:32</a>, <a class="ref" href="euler_2.html#T34" title="EULER_2:th.34">EULER_2:34</a></i>;<br/><a NAME="E5:55_3_1_3_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">q1</font></span>)</span></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c7">p1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">q1</font></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">q1</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>;<br/><a NAME="E6:55_3_1_3_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">q1</font></span>)</span></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c7">p1</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="#E5:55"><i><font color="Green" title="E44">A4</font></i></a>, <a class="txt" href="#E1:55_3_1_3"><i><font color="Green" title="E53">A15</font></i></a>, <a class="txt" href="#E2:55_3_1_3_1_1_2_1_1_1"><i><font color="Green" title="E56">A19</font></i></a>, <a class="ref" href="pepin.html#T11" target="_self" title="PEPIN:th.11">Th11</a>, <a class="ref" href="euler_2.html#T32" title="EULER_2:th.32">EULER_2:32</a></i>;<br/><a NAME="E7:55_3_1_3_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">p1</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="newton.html#T14" title="NEWTON:th.14">NEWTON:14</a></i>;<br/><a NAME="E8:55_3_1_3_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">p1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">p1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">q1</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="#E7:55"><i><font color="Green" title="E46">A6</font></i></a>, <a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></i>;<br/><a NAME="E9:55_3_1_3_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c10">a</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><b>hence </b><a NAME="E10:55_3_1_3_1_1_2_1_1_1"/>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">a</font>
 <b>by </b><i><a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_3_1_3_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:55_3_1_3_1_1_2_1_1_2"/>
<font color="Maroon" title="c6">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">a</font></span><br/><div class="add"><a NAME="E2:55_3_1_3_1_1_2_1_1_2"/><b>then </b>
not <font color="Maroon" title="c6">m</font>,<font color="Maroon" title="c1">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></i>;<br/><a NAME="E3:55_3_1_3_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">m</font> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="#E1:55"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="pepin.html#T2" target="_self" title="PEPIN:th.2">Th2</a></i>;<br/><a NAME="E4:55_3_1_3_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">m</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></i>;<br/><b>hence </b><a NAME="E5:55_3_1_3_1_1_2_1_1_2"/>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">a</font>
 <b>by </b><i><a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</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></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:55_3_1_3_1_1_2"/>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">a</font>
 ; <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="E6:55_3_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">q</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">a</font>
 <b>by </b><i><a class="txt" href="#E1:55"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="#E4:55_3_1_3_1_1"><i><font color="Green" title="E55">A17</font></i></a>, <a class="ref" href="pepin.html#T4" target="_self" title="PEPIN:th.4">Th4</a>, <a class="ref" href="int_2.html#T47" title="INT_2:th.47">INT_2:47</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E8:55_3_1_3_1_1"/><i><font color="Green" title="E56">A20</font></i>: 
<font color="Maroon" title="c6">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">k</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><font color="Maroon" title="c6">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">p1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">q1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">p1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">q1</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">p1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">m</font>
<b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>
;<br/><a NAME="E10:55_3_1_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">p1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c6">m</font>
 <b>by </b><i><a class="txt" href="#E1:55"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="#E8:55_3_1_3_1_1"><i><font color="Green" title="E56">A20</font></i></a>, <a class="ref" href="xcmplx_1.html#T37" title="XCMPLX_1:th.37">XCMPLX_1:37</a></i>;<br/><b>hence </b><a NAME="E11:55_3_1_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">p1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font>
 <b>by </b><i><a class="txt" href="#E2:55"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="nat_d.html#D2" title="NAT_D:def.2">NAT_D:def 2</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>
<b>hence </b><a NAME="E2:55_3_1_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">p1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font>
 ; <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><b>reconsider </b><font color="Maroon" title="c9">t</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">k1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">k2</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default"><a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c3">n</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><font color="Maroon" title="c4">k1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">k2</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">p1</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">t</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="txt" href="#E1:55"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="#E9:55"><i><font color="Green" title="E47">A7</font></i></a>, <a class="txt" href="#E10:55"><i><font color="Green" title="E48">A8</font></i></a>, <a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">p1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1

;<br/><a NAME="E5:55_3_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">k1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">k2</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font>
 <b>by </b><i><a class="txt" href="#E2:55_3_1_3"><i><font color="Green" title="E54">A16</font></i></a></i>;<br/><a NAME="E6:55_3_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c4">k1</font></span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c5">k2</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font>
 <b>by </b><i><a class="ref" href="newton.html#T14" title="NEWTON:th.14">NEWTON:14</a></i>;<br/><b>hence </b><a NAME="E7:55_3_1_3"/>
 <a href="pepin.html#K3" title="PEPIN:func.3">Crypto</a> <span class="p1">(<span class="default"><a href="pepin.html#K3" title="PEPIN:func.3">Crypto</a> <font color="Maroon" title="c6">m</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c4">k1</font></span>)</span>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c5">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font>
 <b>by </b><i><a class="ref" href="pepin.html#T12" target="_self" title="PEPIN:th.12">Th12</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></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E17:55"/>
 <a href="pepin.html#K3" title="PEPIN:func.3">Crypto</a> <span class="p1">(<span class="default"><a href="pepin.html#K3" title="PEPIN:func.3">Crypto</a> <font color="Maroon" title="c6">m</font>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c4">k1</font></span>)</span>,<font color="Maroon" title="c3">n</font>,<font color="Maroon" title="c5">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
