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<div class="add">

<b>let </b><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">n</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="c1">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font> iff <font color="Maroon" title="c1">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</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>for <font color="Olive" title="b1">n</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 <br/> ( $1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive" title="b1">n</font> iff $1 <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive" title="b1">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  );<br/>
<a NAME="E1:19"/><i><font color="Green" title="E10">A1</font></i>: 
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st ( for <font color="Olive" title="b2">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">k</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>]
 
<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" title="c3">m</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> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">m</font>] )</span><br/>

<b>assume </b><a NAME="E1:19_1"/><i><font color="Green" title="E11">A2</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> holds <br/>for <font color="Olive" title="b2">n</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 <br/> ( <font color="Olive" title="b1">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive" title="b2">n</font> iff <font color="Olive" title="b1">k</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Olive" title="b2">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</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="c3">m</font>]</span><br/>

<b>let </b><font color="Maroon" title="c4">n</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="c3">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">n</font> iff <font color="Maroon" title="c3">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  )</span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  implies <font color="Maroon" title="c3">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">n</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:19_1_1"/>
<font color="Maroon" title="c3">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span><br/><b>then consider </b><font color="Maroon" title="c5">k'</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:19_1_1"/><i><font color="Green" title="E12">A3</font></i>: 
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">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/><a NAME="E4:19_1_1"/>
<font color="Maroon" title="c4">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k'</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="pythtrip.html#E3:19_1_1"><i><font color="Green" title="E12">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E5:19_1_1"/>
<font color="Maroon" title="c3">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</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>assume </b><a NAME="E3:19_1"/><i><font color="Green" title="E12">A4</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">n</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1_2"/>
( <font color="Maroon" title="c3">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="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2_1"><b>suppose </b></a><a NAME="E1:19_1_2_1"/>
<font color="Maroon" title="c3">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"> <font color="Maroon" title="c3">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">n</font></span><br/><div class="add"><a NAME="E2:19_1_2_1"/><b>then </b>
<font color="Maroon" title="c4">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <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="pythtrip.html#E3:19_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="int_2.html#T3" title="INT_2:th.3">INT_2:3</a></i>;<br/><a NAME="E3:19_1_2_1"/><b>then </b>
<font color="Maroon" title="c4">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="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></i>;<br/><b>hence </b><a NAME="E4:19_1_2_1"/>
<font color="Maroon" title="c3">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="ref" href="nat_d.html#T6" title="NAT_D:th.6">NAT_D:6</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="19_1_2_2"><b>suppose </b></a><a NAME="E1:19_1_2_2"/>
<font color="Maroon" title="c3">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"> <font color="Maroon" title="c3">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">n</font></span><br/><div class="add"><b>hence </b><a NAME="E2:19_1_2_2"/>
<font color="Maroon" title="c3">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="ref" href="nat_d.html#T6" title="NAT_D:th.6">NAT_D:6</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="19_1_2_3"><b>suppose </b></a><a NAME="E1:19_1_2_3"/><i><font color="Green" title="E13">A5</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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="c3">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">n</font></span><br/><div class="add"><a NAME="E2:19_1_2_3"/><b>then </b>
<font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">p'</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> <b> such that </b><br/><a NAME="E4:19_1_2_3"/><i><font color="Green" title="E14">A6</font></i>: 
( <font color="Maroon" title="c5">p'</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> &amp; <font color="Maroon" title="c5">p'</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">m</font> )
 <b>by </b><i><a class="ref" href="int_2.html#T48" title="INT_2:th.48">INT_2:48</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">p</font> = <font color="Maroon" title="c5">p'</font> as  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <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="pythtrip.html#E4:19_1_2_3"><i><font color="Green" title="E14">A6</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c7">m'</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E7:19_1_2_3"/><i><font color="Green" title="E15">A7</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">m'</font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E4:19_1_2_3"><i><font color="Green" title="E14">A6</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">m'</font> = <font color="Maroon" title="c7">m'</font> 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="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E9:19_1_2_3"/>
<font color="Maroon" title="c3">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c8">m'</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">p</font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E7:19_1_2_3"><i><font color="Green" title="E15">A7</font></i></a></i>;<br/><a NAME="E10:19_1_2_3"/><b>then </b>
<font color="Maroon" title="c6">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><a NAME="E11:19_1_2_3"/><b>then </b>
<font color="Maroon" title="c6">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 <b>by </b><i><a class="txt" href="pythtrip.html#E3:19_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="nat_d.html#T4" title="NAT_D:th.4">NAT_D:4</a></i>;<br/><a NAME="E12:19_1_2_3"/><b>then </b>
<font color="Maroon" title="c6">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="ref" href="newton.html#T98" title="NEWTON:th.98">NEWTON:98</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">n'</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E14:19_1_2_3"/><i><font color="Green" title="E16">A8</font></i>: 
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c9">n'</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>reconsider </b><font color="Maroon" title="c10">n'</font> = <font color="Maroon" title="c9">n'</font> 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="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E16:19_1_2_3"/><i><font color="Green" title="E17">A9</font></i>: 
<font color="Maroon" title="c6">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>by </b><i><a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/><a NAME="E17:19_1_2_3"/><b>then </b><i><font color="Green" title="E18">A10</font></i>: 
<font color="Maroon" title="c6">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">m'</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">m'</font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E1:19_1_2_3"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="pythtrip.html#E7:19_1_2_3"><i><font color="Green" title="E15">A7</font></i></a>, <a class="ref" href="real_2.html#T199" title="REAL_2:th.199">REAL_2:199</a></i>;<br/><b>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="E19:19_1_2_3"/><i><font color="Green" title="E19">A11</font></i>: 
<font color="Maroon" title="c4">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c11">k'</font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E3:19_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><a NAME="E20:19_1_2_3"/><i><font color="Green" title="E20">A12</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n'</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">m'</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c11">k'</font></span>)</span>
 <b>by </b><i><a class="txt" href="pythtrip.html#E7:19_1_2_3"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="pythtrip.html#E14:19_1_2_3"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="pythtrip.html#E19:19_1_2_3"><i><font color="Green" title="E19">A11</font></i></a></i>;<br/><a NAME="E21:19_1_2_3"/>
<font color="Maroon" title="c6">p</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <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="pythtrip.html#E16:19_1_2_3"><i><font color="Green" title="E17">A9</font></i></a>, <a class="ref" href="square_1.html#T74" title="SQUARE_1:th.74">SQUARE_1:74</a></i>;<br/><a NAME="E22:19_1_2_3"/><b>then </b>
<font color="Maroon" title="c10">n'</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">m'</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c11">k'</font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E20:19_1_2_3"><i><font color="Green" title="E20">A12</font></i></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/><a NAME="E23:19_1_2_3"/><b>then </b>
<font color="Maroon" title="c8">m'</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c10">n'</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><a NAME="E24:19_1_2_3"/><b>then </b>
<font color="Maroon" title="c8">m'</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c10">n'</font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E1:19_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="pythtrip.html#E7:19_1_2_3"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="pythtrip.html#E17:19_1_2_3"><i><font color="Green" title="E18">A10</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c12">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E26:19_1_2_3"/><i><font color="Green" title="E21">A13</font></i>: 
<font color="Maroon" title="c10">n'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">m'</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c12">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/><a NAME="E27:19_1_2_3"/>
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">k</font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E7:19_1_2_3"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="pythtrip.html#E14:19_1_2_3"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="pythtrip.html#E26:19_1_2_3"><i><font color="Green" title="E21">A13</font></i></a></i>;<br/><b>hence </b><a NAME="E28:19_1_2_3"/>
<font color="Maroon" title="c3">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</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>
<a NAME="E2:19"/>
for <font color="Olive" title="b1">m</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">m</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S4" title="NAT_1:sch.4">NAT_1:sch 4</a>(<a class="txt" href="pythtrip.html#E1:19"><i><font color="Green" title="E10">A1</font></i></a>);<br/></i>
<b>hence </b><a NAME="E3:19"/>
( <font color="Maroon" title="c1">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font> iff <font color="Maroon" title="c1">m</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
