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

<b>let </b><font color="Maroon" title="c1">p</font> be   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>;<br/><b>let </b><font color="Maroon" title="c2">fp</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:42"/><i><font color="Green" title="E31">A1</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2
 ;<br/>

<b>assume </b><a NAME="E2:42"/><i><font color="Green" title="E32">A2</font></i>: 
for <font color="Olive" title="b1">d</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">fp</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">d</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/>

<a NAME="E3:42"/><b>then </b><i><font color="Green" title="E33">A3</font></i>: 
for <font color="Olive" title="b1">d</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">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">fp</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">d</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
;<br/>
<a NAME="E4:42"/>
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="E5:42"/><b>then </b>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c2">fp</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="int_5.html#E3:42"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="wsierp_1.html#T43" title="WSIERP_1:th.43">WSIERP_1:43</a></i>;<br/>
<a NAME="E6:42"/><b>then </b><i><font color="Green" title="E34">A4</font></i>: 
 <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c2">fp</font></span>)</span>,<font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c2">fp</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</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> 2</span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E1:42"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="int_5.html#T28" target="_self" title="INT_5:th.28">Th28</a></i>;<br/>
<b>set </b><font color="Maroon" title="c3">k</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</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> 2;<br/>
<b>set </b><font color="Maroon" title="c4">f</font> = <font color="Maroon" title="c2">fp</font> <a href="nat_3.html#K3" title="NAT_3:func.3">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</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> 2</span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c2">fp</font> <a href="nat_3.html#K3" title="NAT_3:func.3">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</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> 2</span>)</span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T27" title="FINSEQ_2:th.27">FINSEQ_2:27</a>, <a class="ref" href="numbers.html#T17" title="NUMBERS:th.17">NUMBERS:17</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> =  <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> $1</span>)</span>,<font color="Maroon" title="c1">p</font>;<br/>
<b>consider </b><font color="Maroon" title="c6">fr</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E9:42"/><i><font color="Green" title="E35">A5</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">fr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">fp</font> &amp; ( for <font color="Olive" title="b1">d</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">fr</font> holds <br/><font color="Maroon" title="c6">fr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">d</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S2" title="FINSEQ_1:sch.2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E10:42"/>
for <font color="Olive" title="b1">d</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">fr</font> holds <br/><font color="Maroon" title="c6">fr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:42_1"/>
<font color="Maroon" title="c7">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">fr</font>
 ;<br/>

<a NAME="E2:42_1"/><b>then </b>
<font color="Maroon" title="c6">fr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">d</font></span>)</span>,<font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E9:42"><i><font color="Green" title="E35">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:42_1"/>
<font color="Maroon" title="c6">fr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">fr</font> = <font color="Maroon" title="c6">fr</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T14" title="FINSEQ_2:th.14">FINSEQ_2:14</a></i>;<br/>
<a NAME="E12:42"/><i><font color="Green" title="E36">A6</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">fp</font>
 
<b>by </b><i><a class="ref" href="nat_3.html#D1" title="NAT_3:def.1">NAT_3:def 1</a></i>;<br/>
<a NAME="E13:42"/>
for <font color="Olive" title="b1">d</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">fr</font> holds <br/><font color="Maroon" title="c7">fr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font>,<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:42_2"/><i><font color="Green" title="E37">A7</font></i>: 
<font color="Maroon" title="c8">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">fr</font>
 ;<br/>

<a NAME="E2:42_2"/><b>then </b>
<font color="Maroon" title="c8">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">fp</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E9:42"><i><font color="Green" title="E35">A5</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E3:42_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">d</font></span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">hcf</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="int_5.html#E2:42"><i><font color="Green" title="E32">A2</font></i></a></i>;<br/>
<a NAME="E4:42_2"/><b>then </b>
 <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">d</font></span>)</span>,<font color="Maroon" title="c1">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">d</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</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> 2</span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E1:42"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="int_5.html#T28" target="_self" title="INT_5:th.28">Th28</a></i>;<br/>
<a NAME="E5:42_2"/><b>then </b><i><font color="Green" title="E38">A8</font></i>: 
<font color="Maroon" title="c7">fr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">d</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">d</font></span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</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> 2</span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E9:42"><i><font color="Green" title="E35">A5</font></i></a>, <a class="txt" href="int_5.html#E1:42_2"><i><font color="Green" title="E37">A7</font></i></a></i>;<br/>
<a NAME="E6:42_2"/>
<font color="Maroon" title="c8">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">f</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E9:42"><i><font color="Green" title="E35">A5</font></i></a>, <a class="txt" href="int_5.html#E12:42"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="int_5.html#E1:42_2"><i><font color="Green" title="E37">A7</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<b>hence </b><a NAME="E7:42_2"/>
<font color="Maroon" title="c7">fr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">d</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="int_5.html#E5:42_2"><i><font color="Green" title="E38">A8</font></i></a>, <a class="ref" href="nat_3.html#D1" title="NAT_3:def.1">NAT_3:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:42"/><b>then </b>
 <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c7">fr</font>, <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c5">f</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E9:42"><i><font color="Green" title="E35">A5</font></i></a>, <a class="txt" href="int_5.html#E12:42"><i><font color="Green" title="E36">A6</font></i></a>, <a class="ref" href="int_5.html#T32" target="_self" title="INT_5:th.32">Th32</a></i>;<br/>
<a NAME="E15:42"/><b>then </b><i><font color="Green" title="E37">A9</font></i>: 
 <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c5">f</font>, <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c7">fr</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="ref" href="int_1.html#T35" title="INT_1:th.35">INT_1:35</a></i>;<br/>
<a NAME="E16:42"/>
<font color="Maroon" title="c2">fp</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T27" title="FINSEQ_2:th.27">FINSEQ_2:27</a></i>;<br/>
<a NAME="E17:42"/><b>then </b>
 <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c2">fp</font></span>)</span>,<font color="Maroon" title="c1">p</font>, <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c5">f</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E6:42"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="nat_3.html#T15" title="NAT_3:th.15">NAT_3:15</a></i>;<br/>
<a NAME="E18:42"/><b>then </b>
 <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c2">fp</font></span>)</span>,<font color="Maroon" title="c1">p</font>, <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c7">fr</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E15:42"><i><font color="Green" title="E37">A9</font></i></a>, <a class="ref" href="int_1.html#T36" title="INT_1:th.36">INT_1:36</a></i>;<br/>
<a NAME="E19:42"/><b>then </b><i><font color="Green" title="E38">A10</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c2">fp</font></span>)</span>,<font color="Maroon" title="c1">p</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c7">fr</font></span>)</span>
 
<b>by </b><i><a class="ref" href="int_2.html#T19" title="INT_2:th.19">INT_2:19</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">fr</font>
;<br/>

<a NAME="E20:42"/><i><font color="Green" title="E39">A11</font></i>: 
for <font color="Olive" title="b1">d</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  holds <br/>(  not <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">fr</font> or <font color="Maroon" title="c7">fr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c7">fr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">d</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:42_3"/>
<font color="Maroon" title="c8">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">fr</font>
 ;<br/>

<a NAME="E2:42_3"/><b>then </b>
<font color="Maroon" title="c7">fr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">d</font></span>)</span>,<font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="int_5.html#E9:42"><i><font color="Green" title="E35">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:42_3"/>
( <font color="Maroon" title="c7">fr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c7">fr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 )
 <b>by </b><i><a class="ref" href="int_5.html#T25" target="_self" title="INT_5:th.25">Th25</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_4"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_4"/>
(  <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c2">fp</font></span>)</span>,<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c2">fp</font></span>)</span>,<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 )
 </span><b>by </b><i><a class="ref" href="int_5.html#T25" target="_self" title="INT_5:th.25">Th25</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_4_1"><b>suppose </b></a><a NAME="E1:42_4_1"/><i><font color="Green" title="E40">A12</font></i>: 
 <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c2">fp</font></span>)</span>,<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:42_4_1"/><b>then </b>
 <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c7">fr</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 <b>by </b><i><a class="txt" href="int_5.html#E1:42"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="int_5.html#E19:42"><i><font color="Green" title="E38">A10</font></i></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/><b>hence </b><a NAME="E3:42_4_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">fr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">fp</font> &amp; ( for <font color="Olive" title="b1">d</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">fr</font> holds <br/><font color="Maroon" title="c7">fr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">d</font></span>)</span>,<font color="Maroon" title="c1">p</font> ) &amp;  <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c2">fp</font></span>)</span>,<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c7">fr</font> )
 <b>by </b><i><a class="txt" href="int_5.html#E9:42"><i><font color="Green" title="E35">A5</font></i></a>, <a class="txt" href="int_5.html#E20:42"><i><font color="Green" title="E39">A11</font></i></a>, <a class="txt" href="int_5.html#E1:42_4_1"><i><font color="Green" title="E40">A12</font></i></a>, <a class="ref" href="int_5.html#T31" target="_self" title="INT_5:th.31">Th31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_4_2"><b>suppose </b></a><a NAME="E1:42_4_2"/><i><font color="Green" title="E40">A13</font></i>: 
 <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c2">fp</font></span>)</span>,<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_4_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_4_2_1"/>
 <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c7">fr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E2:42_4_2_1"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 2
 <b>by </b><i><a class="txt" href="int_5.html#E19:42"><i><font color="Green" title="E38">A10</font></i></a>, <a class="txt" href="int_5.html#E1:42_4_2"><i><font color="Green" title="E40">A13</font></i></a></i>;<br/><a NAME="E3:42_4_2_1"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 2
 <b>by </b><i><a class="ref" href="int_2.html#T14" title="INT_2:th.14">INT_2:14</a></i>;<br/><b>hence </b><a NAME="E4:42_4_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="int_5.html#E1:42"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:42_4_2"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">fr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">fp</font> &amp; ( for <font color="Olive" title="b1">d</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">fr</font> holds <br/><font color="Maroon" title="c7">fr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">d</font></span>)</span>,<font color="Maroon" title="c1">p</font> ) &amp;  <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <font color="Maroon" title="c2">fp</font></span>)</span>,<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c7">fr</font> )
 <b>by </b><i><a class="txt" href="int_5.html#E9:42"><i><font color="Green" title="E35">A5</font></i></a>, <a class="txt" href="int_5.html#E20:42"><i><font color="Green" title="E39">A11</font></i></a>, <a class="txt" href="int_5.html#E1:42_4_2"><i><font color="Green" title="E40">A13</font></i></a>, <a class="ref" href="int_5.html#T31" target="_self" title="INT_5:th.31">Th31</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
