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

<b>let </b><font color="Maroon" title="c1">p</font> be   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="newton.html#K13" title="NEWTON:func.13">SetPrimes</a> ] <b>means </b>for <font color="Olive" title="b1">p</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>  st <font color="Olive" title="b1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Product</a> $1 holds <br/><font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> $1;<br/>
<a NAME="E1:13"/><i><font color="Green" title="E7">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">&lt;*&gt;</a> <a href="newton.html#K13" title="NEWTON:func.13">SetPrimes</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">p</font> be   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>;<br/>

<b>assume </b><a NAME="E1:13_1"/>
<font color="Maroon" title="c2">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Product</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">&lt;*&gt;</a> <a href="newton.html#K13" title="NEWTON:func.13">SetPrimes</a> </span>)</span>
 ;<br/>

<a NAME="E2:13_1"/><b>then </b>
<font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a>, <a class="ref" href="rvsum_1.html#T124" title="RVSUM_1:th.124">RVSUM_1:124</a></i>;<br/>
<b>hence </b><a NAME="E3:13_1"/>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">&lt;*&gt;</a> <a href="newton.html#K13" title="NEWTON:func.13">SetPrimes</a> </span>)</span>
 <b>by </b><i><a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E8">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">f</font> be    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="newton.html#K13" title="NEWTON:func.13">SetPrimes</a> ;<br/><b>let </b><font color="Maroon" title="c3">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="newton.html#K13" title="NEWTON:func.13">SetPrimes</a> ;<br/><b>assume </b><a NAME="E1:13_2"/><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">f</font>]
 ;<br/><b>set </b><font color="Maroon" title="c4">f1</font> = <font color="Maroon" title="c2">f</font> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">n</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>;<br/><b>thus </b><a NAME="E2:13_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">f</font> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">n</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">p</font> be   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>;<br/>

<b>assume </b><a NAME="E1:13_2_1"/><i><font color="Green" title="E10">A4</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">n</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c6">ff</font> = <font color="Maroon" title="c2">f</font> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E3:13_2_1"/><i><font color="Green" title="E11">A5</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Product</a> <font color="Maroon" title="c6">ff</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="txt" href="nat_3.html#E1:13_2_1"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="rvsum_1.html#T126" title="RVSUM_1:th.126">RVSUM_1:126</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">nn</font> = <font color="Maroon" title="c3">n</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_2_1_1"/>
( <font color="Maroon" title="c5">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Product</a> <font color="Maroon" title="c2">f</font> or <font color="Maroon" title="c5">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c7">nn</font> )
 </span><b>by </b><i><a class="txt" href="nat_3.html#E3:13_2_1"><i><font color="Green" title="E11">A5</font></i></a>, <a class="ref" href="newton.html#T98" title="NEWTON:th.98">NEWTON:98</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_1_1"><b>suppose </b></a><a NAME="E1:13_2_1_1_1"/>
<font color="Maroon" title="c5">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Product</a> <font color="Maroon" title="c2">f</font>
 ;<br/><div class="add"><a NAME="E2:13_2_1_1_1"/><b>then </b><i><font color="Green" title="E12">A6</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="txt" href="nat_3.html#E1:13_2"><i><font color="Green" title="E9">A3</font></i></a></i>;<br/><a NAME="E3:13_2_1_1_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c2">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">n</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T42" title="FINSEQ_1:th.42">FINSEQ_1:42</a></i>;<br/><b>hence </b><a NAME="E4:13_2_1_1_1"/>
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">n</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="nat_3.html#E2:13_2_1_1_1"><i><font color="Green" title="E12">A6</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_2_1_1_2"><b>suppose </b></a><a NAME="E1:13_2_1_1_2"/><i><font color="Green" title="E12">A7</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c7">nn</font>
 ;<br/><div class="add"><a NAME="E2:13_2_1_1_2"/>
<font color="Maroon" title="c7">nn</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>
 <b>by </b><i><a class="ref" href="newton.html#D6" title="NEWTON:def.6">NEWTON:def 6</a></i>;<br/><a NAME="E3:13_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> )
 <b>by </b><i><a class="txt" href="nat_3.html#E1:13_2_1_1_2"><i><font color="Green" title="E12">A7</font></i></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/><a NAME="E4:13_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">n</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E5:13_2_1_1_2"/><b>then </b><i><font color="Green" title="E13">A8</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">n</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></i>;<br/><a NAME="E6:13_2_1_1_2"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">n</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">n</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T43" title="FINSEQ_1:th.43">FINSEQ_1:43</a></i>;<br/><b>hence </b><a NAME="E7:13_2_1_1_2"/>
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">n</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="nat_3.html#E5:13_2_1_1_2"><i><font color="Green" title="E13">A8</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E3:13"/>
for <font color="Olive" title="b1">p</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="newton.html#K13" title="NEWTON:func.13">SetPrimes</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>]
 
<b>from </b><i><a class="ref" href="finseq_2.html#S2" title="FINSEQ_2:sch.2">FINSEQ_2:sch 2</a>(<a class="txt" href="nat_3.html#E1:13"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="nat_3.html#E2:13"><i><font color="Green" title="E8">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:13"/>
for <font color="Olive" title="b1">f</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="newton.html#K13" title="NEWTON:func.13">SetPrimes</a>   st <font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Product</a> <font color="Olive" title="b1">f</font> holds <br/><font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Olive" title="b1">f</font>
 ;<br/>


</div>
