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

<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">c</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c3">c</font> &amp; <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">c</font> &amp; <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">f</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; ( for <font color="Olive" title="b1">q</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">q</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Olive" title="b1">q</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> holds <br/> ex <font color="Olive" title="b2">a</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 <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b2">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Olive" title="b1">q</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) ) implies <font color="Maroon" title="c1">n</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> )</span><br/>

<b>assume </b><a NAME="E1:25"/><i><font color="Green" title="E26">A1</font></i>: 
( <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c3">c</font> &amp; <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">c</font> &amp; <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">f</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c3">c</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"> (  ex <font color="Olive" title="b1">q</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 <br/>( <font color="Olive" title="b1">q</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Olive" title="b1">q</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> &amp; ( for <font color="Olive" title="b2">a</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 <br/>(  not <span class="p1">(<span class="default"><font color="Olive" title="b2">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Olive" title="b1">q</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) ) ) or <font color="Maroon" title="c1">n</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> )</span><br/>

<b>assume </b><a NAME="E2:25"/><i><font color="Green" title="E27">A2</font></i>: 
for <font color="Olive" title="b1">q</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">q</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Olive" title="b1">q</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> holds <br/> ex <font color="Olive" title="b2">a</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 <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b2">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Olive" title="b1">q</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">n</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a></span><br/>

<b>assume </b><a NAME="E3:25"/><i><font color="Green" title="E28">A3</font></i>: 
not <font color="Maroon" title="c1">n</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:25"/>
<font color="Maroon" title="c2">f</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c3">c</font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:25"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/>
<a NAME="E5:25"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:25"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">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="E7:25"/><i><font color="Green" title="E29">A4</font></i>: 
( <font color="Maroon" title="c4">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c4">p</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c4">p</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> )
 <b>by </b><i><a class="txt" href="nat_4.html#E3:25"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="nat_4.html#E14"><i><font color="Green" title="E14">Lm1</font></i></a></i>;<br/>
<a NAME="E8:25"/><i><font color="Green" title="E30">A5</font></i>: 
for <font color="Olive" title="b1">q1</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">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Olive" title="b1">q1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Maroon" title="c4">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">q1</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">q1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">q1</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c5">q1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c4">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<b>assume </b><a NAME="E1:25_1"/><i><font color="Green" title="E31">A6</font></i>: 
<font color="Maroon" title="c5">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">q1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c4">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<b>assume </b><a NAME="E2:25_1"/><i><font color="Green" title="E32">A7</font></i>: 
<font color="Maroon" title="c5">q1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<a NAME="E3:25_1"/><i><font color="Green" title="E33">A8</font></i>: 
( <font color="Maroon" title="c5">q1</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> &amp; <font color="Maroon" title="c5">q1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">f</font> )
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:25_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="ref" href="nat_3.html#T34" title="NAT_3:th.34">NAT_3:34</a>, <a class="ref" href="nat_3.html#T36" title="NAT_3:th.36">NAT_3:36</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">a</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="E5:25_1"/><i><font color="Green" title="E34">A9</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c5">q1</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="nat_4.html#E2:25"><i><font color="Green" title="E27">A2</font></i></a></i>;<br/>
<a NAME="E6:25_1"/>
<font color="Maroon" title="c5">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E3:25_1"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>
<a NAME="E7:25_1"/><b>then </b><i><font color="Green" title="E35">A10</font></i>: 
<font color="Maroon" title="c5">q1</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:25"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="nat_3.html#D7" title="NAT_3:def.7">NAT_3:def 7</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">n1</font> = <font color="Maroon" title="c5">q1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">f</font>;<br/>
<b>consider </b><font color="Maroon" title="c8">d</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E9:25_1"/><i><font color="Green" title="E36">A11</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">d</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E7:25_1"><i><font color="Green" title="E35">A10</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="c9">d</font> = <font color="Maroon" title="c8">d</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/>
<b>set </b><font color="Maroon" title="c10">k</font> = <font color="Maroon" title="c9">d</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c3">c</font>;<br/>
<a NAME="E11:25_1"/><i><font color="Green" title="E37">A12</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">d</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c3">c</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:25"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="nat_4.html#E9:25_1"><i><font color="Green" title="E36">A11</font></i></a></i>;<br/>
<a NAME="E12:25_1"/><i><font color="Green" title="E38">A13</font></i>: 
<font color="Maroon" title="c9">d</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1_1"/>
<font color="Maroon" title="c9">d</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:25_1_1"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">d</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c3">c</font>
 
;<br/>
<a NAME="E3:25_1_1"/><b>then </b>
<font color="Maroon" title="c9">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:25"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></i>;<br/>
<b>hence </b><a NAME="E4:25_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_4.html#E1:25"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="nat_4.html#E9:25_1"><i><font color="Green" title="E36">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E13:25_1"/><i><font color="Green" title="E39">A14</font></i>: 
<font color="Maroon" title="c5">q1</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:25_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="ref" href="nat_3.html#T34" title="NAT_3:th.34">NAT_3:34</a></i>;<br/>
<a NAME="E14:25_1"/>
not <font color="Maroon" title="c4">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c5">q1</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1_2"/>
<font color="Maroon" title="c4">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c5">q1</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:25_1_2"/><b>then </b>
<font color="Maroon" title="c4">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c5">q1</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="nat_4.html#E7:25"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="newton.html#T63" title="NEWTON:th.63">NEWTON:63</a></i>;<br/>
<b>hence </b><a NAME="E3:25_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_4.html#E7:25"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="nat_4.html#E5:25_1"><i><font color="Green" title="E34">A9</font></i></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E15:25_1"/>
<font color="Maroon" title="c4">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="nat_4.html#E7:25"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="nat_4.html#E2:25_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="txt" href="nat_4.html#E5:25_1"><i><font color="Green" title="E34">A9</font></i></a>, <a class="txt" href="nat_4.html#E11:25_1"><i><font color="Green" title="E37">A12</font></i></a>, <a class="txt" href="nat_4.html#E12:25_1"><i><font color="Green" title="E38">A13</font></i></a>, <a class="txt" href="nat_4.html#E13:25_1"><i><font color="Green" title="E39">A14</font></i></a>, <a class="ref" href="nat_4.html#T23" target="_self" title="NAT_4:th.23">Th23</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E9:25"/><i><font color="Green" title="E31">A15</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E7:25"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E10:25"/><b>then </b><i><font color="Green" title="E32">A16</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <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="nat_4.html#E7:25"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<a NAME="E11:25"/><i><font color="Green" title="E33">A17</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E7:25"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<a NAME="E12:25"/>
for <font color="Olive" title="b1">q1</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">q1</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> holds <br/><font color="Olive" title="b1">q1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">q1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">q1</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">q1</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> implies <font color="Maroon" title="c5">q1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">q1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:25_2"/><i><font color="Green" title="E34">A18</font></i>: 
<font color="Maroon" title="c5">q1</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">q1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">q1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_2_1"/>
( <font color="Maroon" title="c5">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">f</font></span>)</span> or  not <font color="Maroon" title="c5">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_2_1_1"><b>suppose </b></a><a NAME="E1:25_2_1_1"/><i><font color="Green" title="E35">A19</font></i>: 
<font color="Maroon" title="c5">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">q1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">q1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><br/><div class="add"><a NAME="E2:25_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>;<br/><a NAME="E3:25_2_1_1"/><b>then </b>
<font color="Maroon" title="c5">q1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="nat_4.html#E1:25_2"><i><font color="Green" title="E34">A18</font></i></a>, <a class="ref" href="nat_3.html#D8" title="NAT_3:def.8">NAT_3:def 8</a></i>;<br/><a NAME="E4:25_2_1_1"/><b>then </b><i><font color="Green" title="E36">A20</font></i>: 
<font color="Maroon" title="c5">q1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E5:25_2_1_1"/><b>then </b><i><font color="Green" title="E37">A21</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="nat_4.html#E8:25"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="nat_4.html#E1:25_2_1_1"><i><font color="Green" title="E35">A19</font></i></a></i>;<br/><a NAME="E6:25_2_1_1"/><i><font color="Green" title="E38">A22</font></i>: 
<font color="Maroon" title="c5">q1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>by </b><i><a class="txt" href="nat_4.html#E1:25_2"><i><font color="Green" title="E34">A18</font></i></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/><a NAME="E7:25_2_1_1"/><b>then </b>
<font color="Maroon" title="c5">q1</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">f</font></span>)</span> <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="nat_4.html#E4:25_2_1_1"><i><font color="Green" title="E36">A20</font></i></a>, <a class="ref" href="pepin.html#T26" title="PEPIN:th.26">PEPIN:26</a></i>;<br/><a NAME="E8:25_2_1_1"/><b>then </b>
<font color="Maroon" title="c5">q1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">q1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E9:25"><i><font color="Green" title="E31">A15</font></i></a>, <a class="txt" href="nat_4.html#E11:25"><i><font color="Green" title="E33">A17</font></i></a>, <a class="txt" href="nat_4.html#E1:25_2"><i><font color="Green" title="E34">A18</font></i></a>, <a class="txt" href="nat_4.html#E5:25_2_1_1"><i><font color="Green" title="E37">A21</font></i></a>, <a class="ref" href="nat_3.html#T30" title="NAT_3:th.30">NAT_3:30</a>, <a class="ref" href="pepin.html#T8" title="PEPIN:th.8">PEPIN:8</a></i>;<br/><b>hence </b><a NAME="E9:25_2_1_1"/>
<font color="Maroon" title="c5">q1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">q1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E6:25_2_1_1"><i><font color="Green" title="E38">A22</font></i></a>, <a class="ref" href="nat_3.html#T25" title="NAT_3:th.25">NAT_3:25</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="25_2_1_2"><b>suppose </b></a><a NAME="E1:25_2_1_2"/>
not <font color="Maroon" title="c5">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K10" title="POLYNOM1:func.10">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">q1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">q1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><br/><div class="add"><a NAME="E2:25_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">q1</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="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>;<br/><b>hence </b><a NAME="E3:25_2_1_2"/>
<font color="Maroon" title="c5">q1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">q1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:25_2"><i><font color="Green" title="E34">A18</font></i></a>, <a class="ref" href="nat_3.html#D8" title="NAT_3:def.8">NAT_3:def 8</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="E13:25"/><b>then </b>
 ex <font color="Olive" title="b1">n2</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="Maroon" title="c4">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Olive" title="b1">n2</font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:25"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="nat_4.html#E10:25"><i><font color="Green" title="E32">A16</font></i></a>, <a class="ref" href="nat_4.html#T20" target="_self" title="NAT_4:th.20">Th20</a></i>;<br/>
<a NAME="E14:25"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c4">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<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="E15:25"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E10:25"><i><font color="Green" title="E32">A16</font></i></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/>
<a NAME="E16:25"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E11:25"><i><font color="Green" title="E33">A17</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E17:25"/><b>then </b>
<font color="Maroon" title="c4">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E18:25"/><b>then </b>
<font color="Maroon" title="c4">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> <font color="Maroon" title="c2">f</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> 
 
<b>by </b><i><a class="ref" href="square_1.html#T78" title="SQUARE_1:th.78">SQUARE_1:78</a></i>;<br/>
<a NAME="E19:25"/><b>then </b><i><font color="Green" title="E34">A23</font></i>: 
<font color="Maroon" title="c4">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> <font color="Maroon" title="c2">f</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="ref" href="square_1.html#D3" title="SQUARE_1:def.3">SQUARE_1:def 3</a></i>;<br/>
<a NAME="E20:25"/>
<font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">c</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:25"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E21:25"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">f</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></i>;<br/>
<a NAME="E22:25"/><b>then </b><i><font color="Green" title="E35">A24</font></i>: 
<font color="Maroon" title="c4">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> <font color="Maroon" title="c2">f</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:25"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="nat_4.html#E19:25"><i><font color="Green" title="E34">A23</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E23:25"/><i><font color="Green" title="E36">A25</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">c</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:25"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E24:25"/>
<font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:25"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E25:25"/><b>then </b>
<font color="Maroon" title="c3">c</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E26:25"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<b>by </b><i><a class="txt" href="nat_4.html#E23:25"><i><font color="Green" title="E36">A25</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E27:25"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E28:25"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E29:25"/><b>then </b>
<font color="Maroon" title="c4">p</font> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E22:25"><i><font color="Green" title="E35">A24</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E30:25"/><b>then </b>
<font color="Maroon" title="c4">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> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>hence </b><a NAME="E31:25"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_4.html#E7:25"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="square_1.html#D3" title="SQUARE_1:def.3">SQUARE_1: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>
