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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#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<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">m</font>, <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Olive" title="b2">n</font> is <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a> &amp; <font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">n</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a>  holds <br/>( <font color="Olive" title="b1">m</font> is <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a> &amp; <font color="Olive" title="b2">n</font> is <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a> );<br/>
<a NAME="E1:11"/><i><font color="Green" title="E5">A1</font></i>: 
for <font color="Olive" title="b1">mn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st ( for <font color="Olive" title="b2">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   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">mn</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">mn</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">mn</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:11_1"/><i><font color="Green" title="E6">A2</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   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">mn</font> holds <br/>for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Olive" title="b3">n</font> is <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a> &amp; <font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">n</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a>  holds <br/>( <font color="Olive" title="b2">m</font> is <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a> &amp; <font color="Olive" title="b3">n</font> is <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a> )
 ;<br/>

<b>let </b><font color="Maroon" title="c4">m</font>, <font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><a NAME="E2:11_1"/><i><font color="Green" title="E7">A3</font></i>: 
<font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">mn</font>
 ;<br/>

<b>assume </b><a NAME="E3:11_1"/>
<font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c5">n</font> is <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">mn'</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:11_1"/><i><font color="Green" title="E8">A4</font></i>: 
<font color="Maroon" title="c3">mn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">mn'</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 <b>by </b><i><a class="txt" href="pythtrip.html#E2:11_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="pythtrip.html#D3" target="_self" title="PYTHTRIP:def.3">Def3</a></i>;<br/>
<b>assume </b><a NAME="E6:11_1"/><i><font color="Green" title="E9">A5</font></i>: 
<font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c5">n</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 ;<br/>

<a NAME="E7:11_1"/><b>then </b><i><font color="Green" title="E10">A6</font></i>: 
<font color="Maroon" title="c4">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_1_1"/>
( <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c3">mn</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><b>by </b><i><a class="txt" href="pythtrip.html#E2:11_1"><i><font color="Green" title="E7">A3</font></i></a>, <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="11_1_1_1"><b>suppose </b></a><a NAME="E1:11_1_1_1"/><i><font color="Green" title="E11">A7</font></i>: 
<font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_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="E1:11_1_1_1_1_1"/>
( <font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 </span><b>by </b><i><a class="txt" href="pythtrip.html#E1:11_1_1_1"><i><font color="Green" title="E11">A7</font></i></a>, <a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:11_1_1_1_1_1_1"/>
<font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:11_1_1_1_1_1_1"/>
( <font color="Maroon" title="c4">m</font> is <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a> &amp; <font color="Maroon" title="c5">n</font> is <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a> )
 <b>by </b><i><a class="txt" href="pythtrip.html#E7:11_1"><i><font color="Green" title="E10">A6</font></i></a>, <a class="ref" href="newton.html#T65" title="NEWTON:th.65">NEWTON:65</a>, <a class="ref" href="square_1.html#T59" title="SQUARE_1:th.59">SQUARE_1:59</a>, <a class="ref" href="square_1.html#T60" title="SQUARE_1:th.60">SQUARE_1:60</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:11_1_1_1_1_1_2"/>
<font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:11_1_1_1_1_1_2"/>
( <font color="Maroon" title="c4">m</font> is <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a> &amp; <font color="Maroon" title="c5">n</font> is <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a> )
 <b>by </b><i><a class="txt" href="pythtrip.html#E7:11_1"><i><font color="Green" title="E10">A6</font></i></a>, <a class="ref" href="newton.html#T65" title="NEWTON:th.65">NEWTON:65</a>, <a class="ref" href="square_1.html#T59" title="SQUARE_1:th.59">SQUARE_1:59</a>, <a class="ref" href="square_1.html#T60" title="SQUARE_1:th.60">SQUARE_1:60</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_2"><b>suppose </b></a><a NAME="E1:11_1_1_2"/>
<font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:11_1_1_2"/>
( <font color="Maroon" title="c4">m</font> is <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a> &amp; <font color="Maroon" title="c5">n</font> is <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a> )
 <b>by </b><i><a class="ref" href="nat_1.html#T15" title="NAT_1:th.15">NAT_1:15</a>, <a class="ref" href="square_1.html#T59" title="SQUARE_1:th.59">SQUARE_1:59</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_3"><b>suppose </b></a><a NAME="E1:11_1_1_3"/><i><font color="Green" title="E11">A8</font></i>: 
<font color="Maroon" title="c3">mn</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:11_1_1_3"/><b>then </b>
( <font color="Maroon" title="c4">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c5">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="pythtrip.html#E2:11_1"><i><font color="Green" title="E7">A3</font></i></a></i>;<br/><a NAME="E3:11_1_1_3"/><b>then </b><i><font color="Green" title="E12">A9</font></i>: 
( <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 ;<br/><a NAME="E4:11_1_1_3"/>
<font color="Maroon" title="c3">mn</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="pythtrip.html#E1:11_1_1_3"><i><font color="Green" title="E11">A8</font></i></a>, <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="c7">p'</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:11_1_1_3"/><i><font color="Green" title="E13">A10</font></i>: 
( <font color="Maroon" title="c7">p'</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> &amp; <font color="Maroon" title="c7">p'</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">mn</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="c8">p</font> = <font color="Maroon" title="c7">p'</font> as  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="pythtrip.html#E6:11_1_1_3"><i><font color="Green" title="E13">A10</font></i></a></i>;<br/><a NAME="E8:11_1_1_3"/><i><font color="Green" title="E14">A11</font></i>: 
<font color="Maroon" title="c8">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="E9:11_1_1_3"/><b>then </b>
( <font color="Maroon" title="c8">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c8">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 ;<br/><a NAME="E10:11_1_1_3"/><b>then </b>
<font color="Maroon" title="c8">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c8">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c8">p</font>
 <b>by </b><i><a class="ref" href="real_2.html#T144" title="REAL_2:th.144">REAL_2:144</a></i>;<br/><a NAME="E11:11_1_1_3"/><b>then </b><i><font color="Green" title="E15">A12</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c8">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>by </b><i><a class="txt" href="pythtrip.html#E8:11_1_1_3"><i><font color="Green" title="E14">A11</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E12:11_1_1_3"/><i><font color="Green" title="E16">A13</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <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="E13:11_1_1_3"/>
<font color="Maroon" title="c8">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">mn'</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">mn'</font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E5:11_1"><i><font color="Green" title="E8">A4</font></i></a>, <a class="txt" href="pythtrip.html#E6:11_1_1_3"><i><font color="Green" title="E13">A10</font></i></a></i>;<br/><a NAME="E14:11_1_1_3"/><b>then </b>
<font color="Maroon" title="c8">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c6">mn'</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">mn''</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E16:11_1_1_3"/><i><font color="Green" title="E17">A14</font></i>: 
<font color="Maroon" title="c6">mn'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">mn''</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="E17:11_1_1_3"/><i><font color="Green" title="E18">A15</font></i>: 
<font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">mn''</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">mn''</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pythtrip.html#E2:11_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="pythtrip.html#E5:11_1"><i><font color="Green" title="E8">A4</font></i></a>, <a class="txt" href="pythtrip.html#E16:11_1_1_3"><i><font color="Green" title="E17">A14</font></i></a></i>;<br/><div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_1_1_3_1_1"/>
( <font color="Maroon" title="c8">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">m</font> or <font color="Maroon" title="c8">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">n</font> )
 </span><b>by </b><i><a class="txt" href="pythtrip.html#E2:11_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="pythtrip.html#E6:11_1_1_3"><i><font color="Green" title="E13">A10</font></i></a>, <a class="ref" href="newton.html#T98" title="NEWTON:th.98">NEWTON:98</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:11_1_1_3_1_1_1"/><i><font color="Green" title="E19">A16</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">m</font>
 ;<br/><div class="add"><a NAME="E2:11_1_1_3_1_1_1"/><b>then </b><i><font color="Green" title="E20">A17</font></i>: 
not <font color="Maroon" title="c8">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E6:11_1"><i><font color="Green" title="E9">A5</font></i></a>, <a class="ref" href="pythtrip.html#D2" target="_self" title="PYTHTRIP:def.2">Def2</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">m'</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:11_1_1_3_1_1_1"/><i><font color="Green" title="E21">A18</font></i>: 
<font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">m'</font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E1:11_1_1_3_1_1_1"><i><font color="Green" title="E19">A16</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="E5:11_1_1_3_1_1_1"/>
<font color="Maroon" title="c8">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m'</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">mn''</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">mn''</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pythtrip.html#E17:11_1_1_3"><i><font color="Green" title="E18">A15</font></i></a>, <a class="txt" href="pythtrip.html#E4:11_1_1_3_1_1_1"><i><font color="Green" title="E21">A18</font></i></a></i>;<br/><a NAME="E6:11_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">m'</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">mn''</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">mn''</font></span>)</span>
 <b>by </b><i><a class="txt" href="pythtrip.html#E12:11_1_1_3"><i><font color="Green" title="E16">A13</font></i></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/><a NAME="E7:11_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c10">m'</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">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/><a NAME="E8:11_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c10">m'</font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E2:11_1_1_3_1_1_1"><i><font color="Green" title="E20">A17</font></i></a>, <a class="ref" href="newton.html#T98" title="NEWTON:th.98">NEWTON:98</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">m''</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E10:11_1_1_3_1_1_1"/><i><font color="Green" title="E22">A19</font></i>: 
<font color="Maroon" title="c10">m'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">m''</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="c12">m''</font> = <font color="Maroon" title="c11">m''</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E12:11_1_1_3_1_1_1"/><i><font color="Green" title="E23">A20</font></i>: 
<font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c12">m''</font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E4:11_1_1_3_1_1_1"><i><font color="Green" title="E21">A18</font></i></a>, <a class="txt" href="pythtrip.html#E10:11_1_1_3_1_1_1"><i><font color="Green" title="E22">A19</font></i></a></i>;<br/><a NAME="E13:11_1_1_3_1_1_1"/>
<font color="Maroon" title="c12">m''</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="pythtrip.html#E2:11_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="pythtrip.html#E1:11_1_1_3"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="pythtrip.html#E4:11_1_1_3_1_1_1"><i><font color="Green" title="E21">A18</font></i></a>, <a class="txt" href="pythtrip.html#E10:11_1_1_3_1_1_1"><i><font color="Green" title="E22">A19</font></i></a></i>;<br/><a NAME="E14:11_1_1_3_1_1_1"/><b>then </b><i><font color="Green" title="E24">A21</font></i>: 
<font color="Maroon" title="c12">m''</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><a NAME="E15:11_1_1_3_1_1_1"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c12">m''</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">m</font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E11:11_1_1_3"><i><font color="Green" title="E15">A12</font></i></a>, <a class="txt" href="pythtrip.html#E12:11_1_1_3_1_1_1"><i><font color="Green" title="E23">A20</font></i></a>, <a class="ref" href="real_2.html#T199" title="REAL_2:th.199">REAL_2:199</a></i>;<br/><a NAME="E16:11_1_1_3_1_1_1"/><b>then </b><i><font color="Green" title="E25">A22</font></i>: 
<font color="Maroon" title="c12">m''</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">mn</font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E2:11_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="pythtrip.html#E3:11_1_1_3"><i><font color="Green" title="E12">A9</font></i></a>, <a class="txt" href="pythtrip.html#E14:11_1_1_3_1_1_1"><i><font color="Green" title="E24">A21</font></i></a>, <a class="ref" href="real_2.html#T199" title="REAL_2:th.199">REAL_2:199</a></i>;<br/><a NAME="E17:11_1_1_3_1_1_1"/>
<font color="Maroon" title="c8">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">m''</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">mn''</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">mn''</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pythtrip.html#E10:11_1_1_3_1_1_1"><i><font color="Green" title="E22">A19</font></i></a>, <a class="txt" href="pythtrip.html#E17:11_1_1_3"><i><font color="Green" title="E18">A15</font></i></a>, <a class="txt" href="pythtrip.html#E4:11_1_1_3_1_1_1"><i><font color="Green" title="E21">A18</font></i></a></i>;<br/><a NAME="E18:11_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">m''</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">mn''</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">mn''</font></span>)</span>
 <b>by </b><i><a class="txt" href="pythtrip.html#E12:11_1_1_3"><i><font color="Green" title="E16">A13</font></i></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/><a NAME="E19:11_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">m''</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">mn''</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">mn''</font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E12:11_1_1_3"><i><font color="Green" title="E16">A13</font></i></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/><a NAME="E20:11_1_1_3_1_1_1"/><b>then </b><i><font color="Green" title="E26">A23</font></i>: 
<font color="Maroon" title="c12">m''</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">mn''</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 ;<br/><a NAME="E21:11_1_1_3_1_1_1"/>
<font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c12">m''</font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E4:11_1_1_3_1_1_1"><i><font color="Green" title="E21">A18</font></i></a>, <a class="txt" href="pythtrip.html#E10:11_1_1_3_1_1_1"><i><font color="Green" title="E22">A19</font></i></a></i>;<br/><a NAME="E22:11_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">m''</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">m</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="E23:11_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">m''</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="pythtrip.html#E7:11_1"><i><font color="Green" title="E10">A6</font></i></a>, <a class="ref" href="wsierp_1.html#T21" title="WSIERP_1:th.21">WSIERP_1:21</a></i>;<br/><a NAME="E24:11_1_1_3_1_1_1"/><b>then </b><i><font color="Green" title="E27">A24</font></i>: 
<font color="Maroon" title="c12">m''</font>,<font color="Maroon" title="c5">n</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/><a NAME="E25:11_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">m''</font> is <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a>
 <b>by </b><i><a class="txt" href="pythtrip.html#E1:11_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="pythtrip.html#E16:11_1_1_3_1_1_1"><i><font color="Green" title="E25">A22</font></i></a>, <a class="txt" href="pythtrip.html#E20:11_1_1_3_1_1_1"><i><font color="Green" title="E26">A23</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">m'''</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E27:11_1_1_3_1_1_1"/><i><font color="Green" title="E28">A25</font></i>: 
<font color="Maroon" title="c12">m''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">m'''</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 <b>by </b><i><a class="ref" href="pythtrip.html#D3" target="_self" title="PYTHTRIP:def.3">Def3</a></i>;<br/><a NAME="E28:11_1_1_3_1_1_1"/>
<font color="Maroon" title="c4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">m'''</font></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 <b>by </b><i><a class="txt" href="pythtrip.html#E4:11_1_1_3_1_1_1"><i><font color="Green" title="E21">A18</font></i></a>, <a class="txt" href="pythtrip.html#E10:11_1_1_3_1_1_1"><i><font color="Green" title="E22">A19</font></i></a>, <a class="txt" href="pythtrip.html#E27:11_1_1_3_1_1_1"><i><font color="Green" title="E28">A25</font></i></a></i>;<br/><b>hence </b><a NAME="E29:11_1_1_3_1_1_1"/>
( <font color="Maroon" title="c4">m</font> is <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a> &amp; <font color="Maroon" title="c5">n</font> is <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a> )
 <b>by </b><i><a class="txt" href="pythtrip.html#E1:11_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="pythtrip.html#E16:11_1_1_3_1_1_1"><i><font color="Green" title="E25">A22</font></i></a>, <a class="txt" href="pythtrip.html#E20:11_1_1_3_1_1_1"><i><font color="Green" title="E26">A23</font></i></a>, <a class="txt" href="pythtrip.html#E24:11_1_1_3_1_1_1"><i><font color="Green" title="E27">A24</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:11_1_1_3_1_1_2"/><i><font color="Green" title="E19">A26</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">n</font>
 ;<br/><div class="add"><a NAME="E2:11_1_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E20">A27</font></i>: 
not <font color="Maroon" title="c8">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">m</font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E6:11_1"><i><font color="Green" title="E9">A5</font></i></a>, <a class="ref" href="pythtrip.html#D2" target="_self" title="PYTHTRIP:def.2">Def2</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">n'</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:11_1_1_3_1_1_2"/><i><font color="Green" title="E21">A28</font></i>: 
<font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">n'</font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E1:11_1_1_3_1_1_2"><i><font color="Green" title="E19">A26</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="E5:11_1_1_3_1_1_2"/>
<font color="Maroon" title="c8">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">n'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">mn''</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">mn''</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pythtrip.html#E17:11_1_1_3"><i><font color="Green" title="E18">A15</font></i></a>, <a class="txt" href="pythtrip.html#E4:11_1_1_3_1_1_2"><i><font color="Green" title="E21">A28</font></i></a></i>;<br/><a NAME="E6:11_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">n'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">mn''</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">mn''</font></span>)</span>
 <b>by </b><i><a class="txt" href="pythtrip.html#E12:11_1_1_3"><i><font color="Green" title="E16">A13</font></i></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/><a NAME="E7:11_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">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/><a NAME="E8:11_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">p</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#E2:11_1_1_3_1_1_2"><i><font color="Green" title="E20">A27</font></i></a>, <a class="ref" href="newton.html#T98" title="NEWTON:th.98">NEWTON:98</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">n''</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E10:11_1_1_3_1_1_2"/><i><font color="Green" title="E22">A29</font></i>: 
<font color="Maroon" title="c10">n'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">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="c12">n''</font> = <font color="Maroon" title="c11">n''</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E12:11_1_1_3_1_1_2"/><i><font color="Green" title="E23">A30</font></i>: 
<font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c12">n''</font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E4:11_1_1_3_1_1_2"><i><font color="Green" title="E21">A28</font></i></a>, <a class="txt" href="pythtrip.html#E10:11_1_1_3_1_1_2"><i><font color="Green" title="E22">A29</font></i></a></i>;<br/><a NAME="E13:11_1_1_3_1_1_2"/>
<font color="Maroon" title="c12">n''</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="pythtrip.html#E2:11_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="pythtrip.html#E1:11_1_1_3"><i><font color="Green" title="E11">A8</font></i></a>, <a class="txt" href="pythtrip.html#E4:11_1_1_3_1_1_2"><i><font color="Green" title="E21">A28</font></i></a>, <a class="txt" href="pythtrip.html#E10:11_1_1_3_1_1_2"><i><font color="Green" title="E22">A29</font></i></a></i>;<br/><a NAME="E14:11_1_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E24">A31</font></i>: 
<font color="Maroon" title="c12">n''</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><a NAME="E15:11_1_1_3_1_1_2"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c12">n''</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">n</font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E11:11_1_1_3"><i><font color="Green" title="E15">A12</font></i></a>, <a class="txt" href="pythtrip.html#E12:11_1_1_3_1_1_2"><i><font color="Green" title="E23">A30</font></i></a>, <a class="ref" href="real_2.html#T199" title="REAL_2:th.199">REAL_2:199</a></i>;<br/><a NAME="E16:11_1_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E25">A32</font></i>: 
<font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c12">n''</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">mn</font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E2:11_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="pythtrip.html#E3:11_1_1_3"><i><font color="Green" title="E12">A9</font></i></a>, <a class="txt" href="pythtrip.html#E14:11_1_1_3_1_1_2"><i><font color="Green" title="E24">A31</font></i></a>, <a class="ref" href="real_2.html#T199" title="REAL_2:th.199">REAL_2:199</a></i>;<br/><a NAME="E17:11_1_1_3_1_1_2"/>
<font color="Maroon" title="c8">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c12">n''</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">mn''</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">mn''</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pythtrip.html#E10:11_1_1_3_1_1_2"><i><font color="Green" title="E22">A29</font></i></a>, <a class="txt" href="pythtrip.html#E17:11_1_1_3"><i><font color="Green" title="E18">A15</font></i></a>, <a class="txt" href="pythtrip.html#E4:11_1_1_3_1_1_2"><i><font color="Green" title="E21">A28</font></i></a></i>;<br/><a NAME="E18:11_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c12">n''</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">mn''</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">mn''</font></span>)</span>
 <b>by </b><i><a class="txt" href="pythtrip.html#E12:11_1_1_3"><i><font color="Green" title="E16">A13</font></i></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/><a NAME="E19:11_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c12">n''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">mn''</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">mn''</font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E12:11_1_1_3"><i><font color="Green" title="E16">A13</font></i></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/><a NAME="E20:11_1_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E26">A33</font></i>: 
<font color="Maroon" title="c4">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c12">n''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">mn''</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 ;<br/><a NAME="E21:11_1_1_3_1_1_2"/>
<font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c12">n''</font>
 <b>by </b><i><a class="txt" href="pythtrip.html#E4:11_1_1_3_1_1_2"><i><font color="Green" title="E21">A28</font></i></a>, <a class="txt" href="pythtrip.html#E10:11_1_1_3_1_1_2"><i><font color="Green" title="E22">A29</font></i></a></i>;<br/><a NAME="E22:11_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c12">n''</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">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/><a NAME="E23:11_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c12">n''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="pythtrip.html#E7:11_1"><i><font color="Green" title="E10">A6</font></i></a>, <a class="ref" href="wsierp_1.html#T21" title="WSIERP_1:th.21">WSIERP_1:21</a></i>;<br/><a NAME="E24:11_1_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E27">A34</font></i>: 
<font color="Maroon" title="c4">m</font>,<font color="Maroon" title="c12">n''</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/><a NAME="E25:11_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c12">n''</font> is <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a>
 <b>by </b><i><a class="txt" href="pythtrip.html#E1:11_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="pythtrip.html#E16:11_1_1_3_1_1_2"><i><font color="Green" title="E25">A32</font></i></a>, <a class="txt" href="pythtrip.html#E20:11_1_1_3_1_1_2"><i><font color="Green" title="E26">A33</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">n'''</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E27:11_1_1_3_1_1_2"/><i><font color="Green" title="E28">A35</font></i>: 
<font color="Maroon" title="c12">n''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">n'''</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 <b>by </b><i><a class="ref" href="pythtrip.html#D3" target="_self" title="PYTHTRIP:def.3">Def3</a></i>;<br/><a NAME="E28:11_1_1_3_1_1_2"/>
<font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">n'''</font></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 <b>by </b><i><a class="txt" href="pythtrip.html#E4:11_1_1_3_1_1_2"><i><font color="Green" title="E21">A28</font></i></a>, <a class="txt" href="pythtrip.html#E10:11_1_1_3_1_1_2"><i><font color="Green" title="E22">A29</font></i></a>, <a class="txt" href="pythtrip.html#E27:11_1_1_3_1_1_2"><i><font color="Green" title="E28">A35</font></i></a></i>;<br/><b>hence </b><a NAME="E29:11_1_1_3_1_1_2"/>
( <font color="Maroon" title="c4">m</font> is <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a> &amp; <font color="Maroon" title="c5">n</font> is <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a> )
 <b>by </b><i><a class="txt" href="pythtrip.html#E1:11_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="pythtrip.html#E16:11_1_1_3_1_1_2"><i><font color="Green" title="E25">A32</font></i></a>, <a class="txt" href="pythtrip.html#E20:11_1_1_3_1_1_2"><i><font color="Green" title="E26">A33</font></i></a>, <a class="txt" href="pythtrip.html#E24:11_1_1_3_1_1_2"><i><font color="Green" title="E27">A34</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:11"/>
for <font color="Olive" title="b1">mn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">mn</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#E1:11"><i><font color="Green" title="E5">A1</font></i></a>);<br/></i>
<b>hence </b><a NAME="E3:11"/>
( <font color="Maroon" title="c1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font> is <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a> &amp; <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">n</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a>  implies ( <font color="Maroon" title="c1">m</font> is <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a> &amp; <font color="Maroon" title="c2">n</font> is <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a> ) )
 ;<br/>


</div>
