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

<span class="kw">let </span><font color="Maroon" title="c1">h</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> is  <a href="int_6.html#V1" title="INT_6:attr.1">Chinese_Remainder</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">X</font> =  <a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">F</font> =  <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p1">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">m</font> = <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E1:58"/><span class="lab"><font color="Green" title="E48">A1</font></span>: 
<font color="Maroon" title="c1">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> is  <a href="int_6.html#V1" title="INT_6:attr.1">Chinese_Remainder</a> </span><br/>

<a NAME="E2:58"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A2</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number14.html#T44" target="_self" onmouseover="rs('number14/T44')" onmouseout="rh()">Th44</a></span>;<br/>
<a NAME="E3:58"/><span class="lab"><font color="Green" title="E50">A3</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b1">i</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> <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">i</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> <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:58_1"/>
<font color="Maroon" title="c5">i</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> <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> </span><br/>

<a NAME="E2:58_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:58_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="ref" href="newton.html#D6" onmouseover="rs('newton/D6')" onmouseout="rh()">NEWTON:def 6</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:58"/><span class="lab"><font color="Green" title="E51">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" onmouseover="rs('finseq_1/T44')" onmouseout="rh()">FINSEQ_1:44</a></span>;<br/>
<a NAME="E5:58"/><span class="lab"><font color="Green" title="E52">A5</font></span>: 
<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number14.html#T21" target="_self" onmouseover="rs('number14/T21')" onmouseout="rh()">Th21</a></span>;<br/>
<a NAME="E6:58"/><span class="lab"><font color="Green" title="E53">A6</font></span>: 
<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number14.html#T21" target="_self" onmouseover="rs('number14/T21')" onmouseout="rh()">Th21</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">i</font>, <font color="Maroon" title="c6">j</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="int_6.html#D2" onmouseover="rs('int_6/D2')" onmouseout="rh()">INT_6:def 2</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K9" title="XTUPLE_0:func.9">proj1</a> <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> or  not <font color="Maroon" title="c6">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K9" title="XTUPLE_0:func.9">proj1</a> <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> or <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j</font> or <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E7:58"/><span class="lab"><font color="Green" title="E54">A7</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K9" title="XTUPLE_0:func.9">proj1</a> <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> or <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j</font> or <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  )</span><br/>

<a NAME="E8:58"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A8</font></span>: 
<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58"><span class="lab"><font color="Green" title="E50">A3</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E9:58"/><span class="lab"><font color="Green" title="E56">A9</font></span>: 
<font color="Maroon" title="c6">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j</font> or <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  )</span><br/>

<a NAME="E10:58"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A10</font></span>: 
<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58"><span class="lab"><font color="Green" title="E50">A3</font></span></a></span>;<br/>
<a NAME="E11:58"/><span class="lab"><font color="Green" title="E58">A11</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:58"><span class="lab"><font color="Green" title="E56">A9</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E12:58"/><span class="lab"><font color="Green" title="E59">A12</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>;<br/>
<span class="kw">assume </span><a NAME="E13:58"/><span class="lab"><font color="Green" title="E60">A13</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">j</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> </span><br/>

<div><span class="lab"><font color="Green" title="E61">A14</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="58_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 implies ( <font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 &amp; <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  ) )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:58_2"/><span class="lab"><font color="Green" title="E62">A15</font></span>: 
<font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 &amp; <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  )</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">j2</font> = <font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a></span>;<br/><a NAME="E3:58_2"/>
<font color="Maroon" title="c7">j2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_2"><span class="lab"><font color="Green" title="E62">A15</font></span></a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E4:58_2"/><span class="lab"><font color="Green" title="E63">A16</font></span>: 
<font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 &amp; <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  )</span><br/><a NAME="E5:58_2"/>
<font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:58"><span class="lab"><font color="Green" title="E56">A9</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E6:58_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A17</font></span>: 
<font color="Maroon" title="c7">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="txt" href="#E12:58"><span class="lab"><font color="Green" title="E59">A12</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E7:58_2"/>
<font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 &amp; <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  )</span><br/><span class="kw">thus </span><a NAME="E8:58_2"/><span class="lab"><font color="Green" title="E65">A18</font></span>: 
<font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_2"><span class="lab"><font color="Green" title="E63">A16</font></span></a>, <a class="txt" href="#E6:58_2"><span class="lab"><font color="Green" title="E64">A17</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 &amp; <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  )</span><br/><span class="kw">hence </span><a NAME="E9:58_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="number14.html#T40" target="_self" onmouseover="rs('number14/T40')" onmouseout="rh()">Th40</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  )</span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c7">j2</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:58_2"><span class="lab"><font color="Green" title="E65">A18</font></span></a>, <a class="ref" href="number14.html#T23" target="_self" onmouseover="rs('number14/T23')" onmouseout="rh()">Th23</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> </span><br/><span class="kw">thus </span><a NAME="E11:58_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="txt" href="#E8:58_2"><span class="lab"><font color="Green" title="E65">A18</font></span></a>, <a class="ref" href="number14.html#T42" target="_self" onmouseover="rs('number14/T42')" onmouseout="rh()">Th42</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E15:58"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:58"><span class="lab"><font color="Green" title="E54">A7</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_3"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:58_3"/><span class="kw">then </span>
( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="number14.html#E65"><span class="lab"><font color="Green" title="E45">Lm10</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_3_1"><span class="kw">suppose </span></a><a NAME="E1:58_3_1"/><span class="lab"><font color="Green" title="E62">A19</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> </span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_3_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:58_3_1_1"/>
( <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:58"><span class="lab"><font color="Green" title="E58">A11</font></span></a>, <a class="txt" href="number14.html#E65"><span class="lab"><font color="Green" title="E45">Lm10</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:58_3_1_1_1"/>
<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:58_3_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:58"><span class="lab"><font color="Green" title="E60">A13</font></span></a>, <a class="txt" href="#E1:58_3_1"><span class="lab"><font color="Green" title="E62">A19</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:58_3_1_1_2"/>
<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:58_3_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58"><span class="lab"><font color="Green" title="E52">A5</font></span></a>, <a class="txt" href="#E6:58"><span class="lab"><font color="Green" title="E53">A6</font></span></a>, <a class="txt" href="#E8:58"><span class="lab"><font color="Green" title="E55">A8</font></span></a>, <a class="txt" href="#E10:58"><span class="lab"><font color="Green" title="E57">A10</font></span></a>, <a class="txt" href="#E1:58_3_1"><span class="lab"><font color="Green" title="E62">A19</font></span></a>, <a class="ref" href="int_2.html#T30" onmouseover="rs('int_2/T30')" onmouseout="rh()">INT_2:30</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_3_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:58_3_1_1_3"/><span class="lab"><font color="Green" title="E63">A20</font></span>: 
<font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> </span><br/><div class="add"><a NAME="E2:58_3_1_1_3"/><span class="kw">then </span>
( 2,<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  or 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:58"><span class="lab"><font color="Green" title="E61">A14</font></span></a>, <a class="ref" href="int_2.html#T30" onmouseover="rs('int_2/T30')" onmouseout="rh()">INT_2:30</a>, <a class="ref" href="xprimes1.html#T2" onmouseover="rs('xprimes1/T2')" onmouseout="rh()">XPRIMES1:2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:58_3_1_1_3"/>
<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:58"><span class="lab"><font color="Green" title="E61">A14</font></span></a>, <a class="txt" href="#E1:58_3_1"><span class="lab"><font color="Green" title="E62">A19</font></span></a>, <a class="txt" href="#E1:58_3_1_1_3"><span class="lab"><font color="Green" title="E63">A20</font></span></a>, <a class="ref" href="number14.html#T21" target="_self" onmouseover="rs('number14/T21')" onmouseout="rh()">Th21</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_3_2"><span class="kw">suppose </span></a><a NAME="E1:58_3_2"/><span class="lab"><font color="Green" title="E62">A21</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> </span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_3_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:58_3_2_1"/>
( <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:58"><span class="lab"><font color="Green" title="E58">A11</font></span></a>, <a class="txt" href="number14.html#E65"><span class="lab"><font color="Green" title="E45">Lm10</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_3_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:58_3_2_1_1"/>
<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:58_3_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:58"><span class="lab"><font color="Green" title="E52">A5</font></span></a>, <a class="txt" href="#E6:58"><span class="lab"><font color="Green" title="E53">A6</font></span></a>, <a class="txt" href="#E1:58_3_2"><span class="lab"><font color="Green" title="E62">A21</font></span></a>, <a class="ref" href="int_2.html#T30" onmouseover="rs('int_2/T30')" onmouseout="rh()">INT_2:30</a>, <a class="ref" href="xprimes1.html#T2" onmouseover="rs('xprimes1/T2')" onmouseout="rh()">XPRIMES1:2</a>, <a class="ref" href="xprimes1.html#T3" onmouseover="rs('xprimes1/T3')" onmouseout="rh()">XPRIMES1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_3_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:58_3_2_1_2"/>
<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:58_3_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:58"><span class="lab"><font color="Green" title="E60">A13</font></span></a>, <a class="txt" href="#E1:58_3_2"><span class="lab"><font color="Green" title="E62">A21</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_3_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:58_3_2_1_3"/>
<font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:58_3_2_1_3"/>
<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:58"><span class="lab"><font color="Green" title="E53">A6</font></span></a>, <a class="txt" href="#E14:58"><span class="lab"><font color="Green" title="E61">A14</font></span></a>, <a class="txt" href="#E1:58_3_2"><span class="lab"><font color="Green" title="E62">A21</font></span></a>, <a class="ref" href="int_2.html#T30" onmouseover="rs('int_2/T30')" onmouseout="rh()">INT_2:30</a>, <a class="ref" href="xprimes1.html#T3" onmouseover="rs('xprimes1/T3')" onmouseout="rh()">XPRIMES1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_3_3"><span class="kw">suppose </span></a><a NAME="E1:58_3_3"/><span class="lab"><font color="Green" title="E62">A22</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">i2</font> = <font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a></span>;<br/><a NAME="E3:58_3_3"/>
<font color="Maroon" title="c7">i2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_3_3"><span class="lab"><font color="Green" title="E62">A22</font></span></a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E4:58_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A23</font></span>: 
<font color="Maroon" title="c7">i2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E5:58_3_3"/>
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:58"><span class="lab"><font color="Green" title="E54">A7</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E6:58_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A24</font></span>: 
<font color="Maroon" title="c7">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="txt" href="#E12:58"><span class="lab"><font color="Green" title="E59">A12</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/><a NAME="E7:58_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A25</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_3_3"><span class="lab"><font color="Green" title="E63">A23</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E8:58_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A26</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="number14.html#T40" target="_self" onmouseover="rs('number14/T40')" onmouseout="rh()">Th40</a></span>;<br/><span class="lab"><font color="Green" title="E67">A27</font></span>: <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c7">i2</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:58_3_3"><span class="lab"><font color="Green" title="E65">A25</font></span></a>, <a class="ref" href="number14.html#T23" target="_self" onmouseover="rs('number14/T23')" onmouseout="rh()">Th23</a></span>
;<br/><a NAME="E10:58_3_3"/><span class="lab"><font color="Green" title="E68">A28</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="txt" href="#E7:58_3_3"><span class="lab"><font color="Green" title="E65">A25</font></span></a>, <a class="ref" href="number14.html#T42" target="_self" onmouseover="rs('number14/T42')" onmouseout="rh()">Th42</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_3_3_2"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:58_3_3_2"/>
( <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:58"><span class="lab"><font color="Green" title="E58">A11</font></span></a>, <a class="txt" href="number14.html#E65"><span class="lab"><font color="Green" title="E45">Lm10</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_3_3_2_1"><span class="kw">suppose </span></a><a NAME="E1:58_3_3_2_1"/><span class="lab"><font color="Green" title="E69">A29</font></span>: 
<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> </span><br/><div class="add"><a NAME="E2:58_3_3_2_1"/>
( 2,<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  or 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:58_3_3"><span class="lab"><font color="Green" title="E68">A28</font></span></a>, <a class="ref" href="int_2.html#T30" onmouseover="rs('int_2/T30')" onmouseout="rh()">INT_2:30</a>, <a class="ref" href="xprimes1.html#T2" onmouseover="rs('xprimes1/T2')" onmouseout="rh()">XPRIMES1:2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:58_3_3_2_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="txt" href="#E7:58_3_3"><span class="lab"><font color="Green" title="E65">A25</font></span></a>, <a class="txt" href="#E9:58_3_3"><span class="lab"><font color="Green" title="E67">A27</font></span></a>, <a class="txt" href="#E1:58_3_3_2_1"><span class="lab"><font color="Green" title="E69">A29</font></span></a>, <a class="ref" href="number14.html#T40" target="_self" onmouseover="rs('number14/T40')" onmouseout="rh()">Th40</a>, <a class="ref" href="number14.html#T21" target="_self" onmouseover="rs('number14/T21')" onmouseout="rh()">Th21</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_3_3_2_2"><span class="kw">suppose </span></a><a NAME="E1:58_3_3_2_2"/>
<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:58_3_3_2_2"/>
<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:58"><span class="lab"><font color="Green" title="E53">A6</font></span></a>, <a class="txt" href="#E8:58_3_3"><span class="lab"><font color="Green" title="E66">A26</font></span></a>, <a class="txt" href="#E9:58_3_3"><span class="lab"><font color="Green" title="E67">A27</font></span></a>, <a class="txt" href="#E10:58_3_3"><span class="lab"><font color="Green" title="E68">A28</font></span></a>, <a class="ref" href="int_2.html#T30" onmouseover="rs('int_2/T30')" onmouseout="rh()">INT_2:30</a>, <a class="ref" href="xprimes1.html#T3" onmouseover="rs('xprimes1/T3')" onmouseout="rh()">XPRIMES1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_3_3_2_3"><span class="kw">suppose </span></a><a NAME="E1:58_3_3_2_3"/><span class="lab"><font color="Green" title="E69">A30</font></span>: 
<font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">j2</font> = <font color="Maroon" title="c6">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:58"><span class="lab"><font color="Green" title="E61">A14</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_3_3_2_3_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:58_3_3_2_3_1"/>
( <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">j</font> or <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">j</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:58"><span class="lab"><font color="Green" title="E60">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_3_3_2_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:58_3_3_2_3_1_1"/>
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">j</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> </span><br/><div class="add"><a NAME="E2:58_3_3_2_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">i2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E3:58_3_3_2_3_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="txt" href="#E14:58"><span class="lab"><font color="Green" title="E61">A14</font></span></a>, <a class="txt" href="#E4:58_3_3"><span class="lab"><font color="Green" title="E63">A23</font></span></a>, <a class="txt" href="#E1:58_3_3_2_3"><span class="lab"><font color="Green" title="E69">A30</font></span></a>, <a class="ref" href="finseq_1.html#D14" onmouseover="rs('finseq_1/D14')" onmouseout="rh()">FINSEQ_1:def 14</a></span>;<br/><span class="kw">hence </span><a NAME="E4:58_3_3_2_3_1_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:58"><span class="lab"><font color="Green" title="E55">A8</font></span></a>, <a class="txt" href="#E14:58"><span class="lab"><font color="Green" title="E61">A14</font></span></a>, <a class="txt" href="#E9:58_3_3"><span class="lab"><font color="Green" title="E67">A27</font></span></a>, <a class="txt" href="#E1:58_3_3_2_3"><span class="lab"><font color="Green" title="E69">A30</font></span></a>, <a class="ref" href="int_2.html#T30" onmouseover="rs('int_2/T30')" onmouseout="rh()">INT_2:30</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_3_3_2_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:58_3_3_2_3_1_2"/>
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">j</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> </span><br/><div class="add"><a NAME="E2:58_3_3_2_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">i2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c8">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E3:58_3_3_2_3_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="txt" href="#E14:58"><span class="lab"><font color="Green" title="E61">A14</font></span></a>, <a class="txt" href="#E6:58_3_3"><span class="lab"><font color="Green" title="E64">A24</font></span></a>, <a class="txt" href="#E1:58_3_3_2_3"><span class="lab"><font color="Green" title="E69">A30</font></span></a>, <a class="ref" href="finseq_1.html#D14" onmouseover="rs('finseq_1/D14')" onmouseout="rh()">FINSEQ_1:def 14</a></span>;<br/><span class="kw">hence </span><a NAME="E4:58_3_3_2_3_1_2"/>
<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c1">h</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:58"><span class="lab"><font color="Green" title="E55">A8</font></span></a>, <a class="txt" href="#E14:58"><span class="lab"><font color="Green" title="E61">A14</font></span></a>, <a class="txt" href="#E9:58_3_3"><span class="lab"><font color="Green" title="E67">A27</font></span></a>, <a class="txt" href="#E1:58_3_3_2_3"><span class="lab"><font color="Green" title="E69">A30</font></span></a>, <a class="ref" href="int_2.html#T30" onmouseover="rs('int_2/T30')" onmouseout="rh()">INT_2:30</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
