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<div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="newton.html#NM1">Prime</a>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="trees_4.html#M1">FinSequence</a> of  <a href="newton.html#K13">SetPrimes</a> ] <b>means </b>for <font color="Olive">p</font> being  <a href="newton.html#NM1">Prime</a>  st <font color="Olive">p</font> <a href="nat_1.html#R1">divides</a>  <a href="wsierp_1.html#K8">Product</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:13"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="wsierp_1.html#K5">&lt;*&gt;</a> <a href="newton.html#K13">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">p</font> be   <a href="newton.html#NM1">Prime</a>;<br/>

<b>assume </b><a NAME="E1:13_1"/>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a>  <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K5">&lt;*&gt;</a> <a href="newton.html#K13">SetPrimes</a> </span>)</span>
 ;<br/>

<a NAME="E2:13_1"/><b>then </b>
<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T54">NAT_1:54</a>, <a class="ref" href="rvsum_1.html#T124">RVSUM_1:124</a></i>;<br/>
<b>hence </b><a NAME="E3:13_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K5">&lt;*&gt;</a> <a href="newton.html#K13">SetPrimes</a> </span>)</span>
 <b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E43</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">f</font> be    <a href="trees_4.html#M1">FinSequence</a> of  <a href="newton.html#K13">SetPrimes</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="newton.html#K13">SetPrimes</a> ;<br/><b>assume </b><a NAME="E1:13_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">f</font>]
 ;<br/><b>set </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> <a href="wsierp_1.html#K4">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="binarith.html#K11">*&gt;</a></span>;<br/><b>thus </b><a NAME="E2:13_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">f</font> <a href="wsierp_1.html#K4">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="binarith.html#K11">*&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">p</font> be   <a href="newton.html#NM1">Prime</a>;<br/>

<b>assume </b><a NAME="E1:13_2_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a>  <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">ff</font> = <font color="Maroon">f</font> as    <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E3:13_2_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <font color="Maroon">ff</font></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="rvsum_1.html#T126">RVSUM_1:126</a></i>;<br/>
<b>reconsider </b><font color="Maroon">nn</font> = <font color="Maroon">n</font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<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">p</font> <a href="nat_1.html#R1">divides</a>  <a href="wsierp_1.html#K8">Product</a> <font color="Maroon">f</font> or <font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">nn</font> )
 </span><b>by </b><i>, <a class="ref" href="newton.html#T98">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">p</font> <a href="nat_1.html#R1">divides</a>  <a href="wsierp_1.html#K8">Product</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E2:13_2_1_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/><a NAME="E3:13_2_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T42">FINSEQ_1:42</a></i>;<br/><b>hence </b><a NAME="E4:13_2_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
 <b>by </b><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">E52</font></i>: 
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">nn</font>
 ;<br/><div class="add"><a NAME="E2:13_2_1_1_2"/>
<font color="Maroon">nn</font> is <a href="int_2.html#V1">prime</a>
 <b>by </b><i><a class="ref" href="newton.html#D6">NEWTON:def 6</a></i>;<br/><a NAME="E3:13_2_1_1_2"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/><a NAME="E4:13_2_1_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">n</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:13_2_1_1_2"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/><a NAME="E6:13_2_1_1_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="binarith.html#K11">*&gt;</a></span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T43">FINSEQ_1:43</a></i>;<br/><b>hence </b><a NAME="E7:13_2_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
 <b>by </b><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">p</font> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="newton.html#K13">SetPrimes</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>]
 
<b>from </b><i><a class="ref" href="finseq_2.html#S2">FINSEQ_2:sch 2</a>(, );<br/></i>
<b>hence </b><a NAME="E4:13"/>
for <font color="Olive">f</font> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="newton.html#K13">SetPrimes</a>   st <font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a>  <a href="wsierp_1.html#K8">Product</a> <font color="Olive">f</font> holds <br/><font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">f</font>
 ;<br/>


</div>
