<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">a</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">b</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:101"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="int_2.html#R2">are_relative_prime</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">an</font> = <font color="Maroon">a</font>, <font color="Maroon">bn</font> = <font color="Maroon">b</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:101_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="newton.html#K13">SetPrimes</a> 
 ;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">i</font> as   <a href="newton.html#NM1">Prime</a> <b>by </b><i><a class="ref" href="nat_3.html#T3" target="_self">Th3</a>, <a class="ref" href="newton.html#D6">NEWTON:def 6</a></i>;<br/><a NAME="E3:101_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">an</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">bn</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E4:101_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/><a NAME="E5:101_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:101_1_1"/>
( <font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_1"><b>suppose </b></a><a NAME="E1:101_1_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:101_1_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
( <font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="nat_3.html#T4" target="_self">Th4</a>, <a class="ref" href="nat_1.html#T23">NAT_1:23</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font> = 
0
<b>by </b><i>, <a class="txt" href="nat_3.html#E5:101_1"><i><font color="Green">E50</font></i></a></i>
<br/>.= 
0 <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="nat_3.html#T5" target="_self">Th5</a>, <a class="txt" href="nat_3.html#E5:101_1"><i><font color="Green">E50</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">a</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="nat_3.html#T5" target="_self">Th5</a>, <a class="txt" href="nat_3.html#E5:101_1"><i><font color="Green">E50</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">a</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="polynom1.html#D5">POLYNOM1:def 5</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_2"><b>suppose </b></a><a NAME="E1:101_1_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>thus </b><a NAME="E2:101_1_1_2"/>
<span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">a</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_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="E1:101_1_1_2_1_1"/>
( <font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span><b>by </b><i><a class="ref" href="nat_3.html#T4" target="_self">Th4</a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:101_1_1_2_1_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><div><i><font color="Green">E74</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:101_1_1_2_1_1_1_1"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><b>then consider </b><font color="Maroon">kb</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:101_1_1_2_1_1_1_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">kb</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><a NAME="E4:101_1_1_2_1_1_1_1"/>
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:101_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="newton.html#K3">|^</a> <font color="Maroon">kb</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/><b>then consider </b><font color="Maroon">lb</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:101_1_1_2_1_1_1_1"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="newton.html#K3">|^</a> <font color="Maroon">kb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">lb</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E8:101_1_1_2_1_1_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="newton.html#K3">|^</a> <font color="Maroon">kb</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">lb</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E9:101_1_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><b>consider </b><font color="Maroon">ka</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E11:101_1_1_2_1_1_1_1"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ka</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_3.html#T5" target="_self">Th5</a>, <a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><a NAME="E12:101_1_1_2_1_1_1_1"/>
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/><a NAME="E13:101_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="newton.html#K3">|^</a> <font color="Maroon">ka</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/><b>then consider </b><font color="Maroon">la</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E15:101_1_1_2_1_1_1_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="newton.html#K3">|^</a> <font color="Maroon">ka</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">la</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E16:101_1_1_2_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="newton.html#K3">|^</a> <font color="Maroon">ka</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">la</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_3.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E17:101_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E18:101_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> 1
 <b>by </b><i>, , <a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E19:101_1_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/></div><b>end;</b></div><b>hence </b><span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="nat_3.html#T4" target="_self">Th4</a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">a</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K23">+</a> 0
<b>by </b><i><a class="ref" href="nat_3.html#T5" target="_self">Th5</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">a</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="ref" href="nat_3.html#T6" target="_self">Th6</a>, <a class="txt" href="nat_3.html#E5:101_1"><i><font color="Green">E50</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">a</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="polynom1.html#D5">POLYNOM1:def 5</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:101_1_1_2_1_1_2"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><div><i><font color="Green">E109</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:101_1_1_2_1_1_2_1"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><b>consider </b><font color="Maroon">kb</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:101_1_1_2_1_1_2_1"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">kb</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_3.html#T5" target="_self">Th5</a>, <a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><a NAME="E4:101_1_1_2_1_1_2_1"/>
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:101_1_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="newton.html#K3">|^</a> <font color="Maroon">kb</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/><b>then consider </b><font color="Maroon">lb</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:101_1_1_2_1_1_2_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="newton.html#K3">|^</a> <font color="Maroon">kb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">lb</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E8:101_1_1_2_1_1_2_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="newton.html#K3">|^</a> <font color="Maroon">kb</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">lb</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E9:101_1_1_2_1_1_2_1"/><b>then </b><i><font color="Green">E113</font></i>: 
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><b>consider </b><font color="Maroon">ka</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E11:101_1_1_2_1_1_2_1"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ka</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><a NAME="E12:101_1_1_2_1_1_2_1"/>
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/><a NAME="E13:101_1_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="newton.html#K3">|^</a> <font color="Maroon">ka</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="nat_3.html#D1" target="_self">Def1</a>, <a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/><b>then consider </b><font color="Maroon">la</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E15:101_1_1_2_1_1_2_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="newton.html#K3">|^</a> <font color="Maroon">ka</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">la</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E16:101_1_1_2_1_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="newton.html#K3">|^</a> <font color="Maroon">ka</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">la</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E17:101_1_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E18:101_1_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> 1
 <b>by </b><i>, , <a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E19:101_1_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/></div><b>end;</b></div><b>hence </b><span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="nat_3.html#T4" target="_self">Th4</a>, , </i>
<br/>.= 
0 <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="ref" href="nat_3.html#T5" target="_self">Th5</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">a</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="ref" href="nat_3.html#T6" target="_self">Th6</a>, <a class="txt" href="nat_3.html#E5:101_1"><i><font color="Green">E50</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">a</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="polynom1.html#D5">POLYNOM1:def 5</a></i>
;<br/><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>
<b>hence </b><a NAME="E4:101"/>
 <a href="nat_3.html#NK15" target="_self">ppf</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">a</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


</div>
