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

<div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><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/><a NAME="E1:51_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E2:51_1"/>
<font color="Maroon">F</font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/></div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">a</font>, <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">F</font> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="newton.html#K13">SetPrimes</a> <br/> for <font color="Olive">p</font> being  <a href="newton.html#NM1">Prime</a>  st <font color="Olive">X</font> <a href="tarski.html#R1">c=</a>  <a href="newton.html#K13">SetPrimes</a>  &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Olive">k</font> &amp; <font color="Olive">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K14">Sgm</a> <font color="Olive">X</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K8">Product</a> <font color="Olive">F</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</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="relset_1.html#K5">rng</a> <font color="Olive">F</font> implies <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> 1 ) &amp; ( not <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Olive">F</font> implies <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> 0 ) );<br/>
<a NAME="E2:51"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><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">p</font> be   <a href="newton.html#NM1">Prime</a>;<br/>







<b>assume </b><a NAME="E1:51_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="newton.html#K13">SetPrimes</a>  &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font> &amp; <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K8">Product</a> <font color="Maroon">F</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<b>assume </b><a NAME="E2:51_2"/><i><font color="Green">E46</font></i>: 
( ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> &amp; not <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1 ) or ( not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> &amp; not <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 ) )
 ;<br/>

<a NAME="E3:51_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:51_2_1"/>
( ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> &amp; not <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1 ) or ( not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> &amp; not <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_1_1"><b>suppose </b></a><a NAME="E1:51_2_1_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> &amp; not <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:51_2_1_1"/>
 <a href="relat_1.html#K2">rng</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>hence </b><a NAME="E3:51_2_1_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="nat_4.html#E3:51"><i><font color="Green">E54</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_1_2"><b>suppose </b></a><a NAME="E1:51_2_1_2"/><i><font color="Green">E53</font></i>: 
( not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> &amp; not <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><a NAME="E2:51_2_1_2"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a>, <a class="ref" href="rvsum_1.html#T124">RVSUM_1:124</a>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/><b>hence </b><a NAME="E3:51_2_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_3.html#T21">NAT_3:21</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:51"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:51_3"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<a NAME="E2:51_3"/>
for <font color="Olive">a</font>, <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">F</font> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="newton.html#K13">SetPrimes</a> <br/> for <font color="Olive">p</font> being  <a href="newton.html#NM1">Prime</a>  st <font color="Olive">X</font> <a href="tarski.html#R1">c=</a>  <a href="newton.html#K13">SetPrimes</a>  &amp; <font color="Olive">X</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Olive">k</font> &amp; <font color="Olive">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K14">Sgm</a> <font color="Olive">X</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K8">Product</a> <font color="Olive">F</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 holds <br/>( ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Olive">F</font> implies <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> 1 ) &amp; ( not <font color="Olive">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Olive">F</font> implies <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> 0 ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><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">p</font> be   <a href="newton.html#NM1">Prime</a>;<br/>







<b>assume </b><a NAME="E1:51_3_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="newton.html#K13">SetPrimes</a>  &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font> &amp; <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K8">Product</a> <font color="Maroon">F</font> )
 ;<br/>

<b>assume </b><a NAME="E2:51_3_1"/><i><font color="Green">E58</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 ;<br/>

<b>set </b><font color="Maroon">x</font> = <font color="Maroon">F</font> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>;<br/>
<a NAME="E3:51_3_1"/><i><font color="Green">E60</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:51_3"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:51_3_1"/><b>then </b>
<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E5:51_3_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">F</font> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:51_3_1"/><i><font color="Green">E62</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="newton.html#K13">SetPrimes</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">F</font> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> as   <a href="newton.html#NM1">Prime</a> <b>by </b><i>, <a class="txt" href="nat_4.html#E2:51_3_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="newton.html#D6">NEWTON:def 6</a></i>;<br/>
<b>set </b><font color="Maroon">X1</font> = <font color="Maroon">X</font> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>;<br/>
<a NAME="E8:51_3_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">X</font> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E9:51_3_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">X</font> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="newton.html#K13">SetPrimes</a> 
 
<b>by </b><i><a class="txt" href="nat_4.html#E3:51"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E10:51_3_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">X</font> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E3:51"><i><font color="Green">E54</font></i></a>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>set </b><font color="Maroon">F1</font> =  <a href="finseq_1.html#K14">Sgm</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K4">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">F1</font> =  <a href="finseq_1.html#K14">Sgm</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K4">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> as    <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>set </b><font color="Maroon">a1</font> =  <a href="wsierp_1.html#K8">Product</a> <font color="Maroon">F1</font>;<br/>
<b>set </b><font color="Maroon">F2</font> =  <a href="finseq_1.html#K14">Sgm</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>;<br/>
<a NAME="E12:51_3_1"/><i><font color="Green">E68</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E3:51"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:51_3_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:51_3_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:51_3_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E3:51"><i><font color="Green">E54</font></i></a>, , <a class="txt" href="nat_4.html#E6:51_3_1"><i><font color="Green">E62</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:51_3_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E15:51_3_1"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">m1</font>, <font color="Olive">n1</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">n1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> holds <br/><font color="Olive">m1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">n1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:51_3_1_2"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">m1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">n1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 ;<br/>

<a NAME="E2:51_3_1_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">m1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E6:51_3_1"><i><font color="Green">E62</font></i></a>, , </i>;<br/>
<a NAME="E3:51_3_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">F</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="nat_4.html#E3:51"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_3.html#T99">FINSEQ_3:99</a></i>;<br/>
<a NAME="E4:51_3_1_2"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">m1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K2">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="nat_4.html#E10:51_3_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<b>set </b><font color="Maroon">l</font> = <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font>;<br/>
<a NAME="E5:51_3_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</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="funct_1.html#K2">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="nat_4.html#E12:51_3_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:51_3_1_2"/><b>then </b><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:51_3_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="funct_1.html#T55">FUNCT_1:55</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">l</font> = <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="nat_4.html#E14:51_3_1"><i><font color="Green">E69</font></i></a></i>;<br/>
<a NAME="E8:51_3_1_2"/><i><font color="Green">E76</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 )
 
<b>by </b><i><a class="txt" href="nat_4.html#E14:51_3_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>set </b><font color="Maroon">n'</font> = <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1;<br/>
<a NAME="E9:51_3_1_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">l</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T57">FUNCT_1:57</a>, <a class="txt" href="nat_4.html#E6:51_3_1"><i><font color="Green">E62</font></i></a>, , , </i>;<br/>
<a NAME="E10:51_3_1_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E11:51_3_1_2"/>
not <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">l</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a>, , <a class="txt" href="nat_4.html#E1:51_3_1_2"><i><font color="Green">E71</font></i></a></i>;<br/>
<a NAME="E12:51_3_1_2"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E15:51_3_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/>
<b>thus </b><a NAME="E13:51_3_1_2"/>
<font color="Maroon">m1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E3:51"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a>, <a class="txt" href="nat_4.html#E15:51_3_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="nat_4.html#E1:51_3"><i><font color="Green">E55</font></i></a>, <a class="txt" href="nat_4.html#E1:51_3_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="nat_4.html#E2:51_3_1_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="nat_4.html#E3:51_3_1_2"><i><font color="Green">E73</font></i></a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E80</font></i>: <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K4">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> = 
<font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
<br/>.= 
<font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a>, , <a class="txt" href="nat_4.html#E6:51_3_1"><i><font color="Green">E62</font></i></a></i>
;<br/>
<a NAME="E17:51_3_1"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="wsierp_1.html#K4">^</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="nat_4.html#E3:51"><i><font color="Green">E54</font></i></a>, <a class="txt" href="nat_4.html#E5:51_3_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="nat_4.html#E9:51_3_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="nat_4.html#E8:51_3_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="finseq_3.html#T48">FINSEQ_3:48</a></i>;<br/>
<a NAME="E18:51_3_1"/><b>then </b><i><font color="Green">E82</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E19:51_3_1"/><i><font color="Green">E83</font></i>: 
 <a href="finseq_1.html#K14">Sgm</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T50">FINSEQ_3:50</a></i>;<br/>
<a NAME="E20:51_3_1"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i>, <a class="txt" href="nat_4.html#E1:51_3"><i><font color="Green">E55</font></i></a>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E21:51_3_1"/><i><font color="Green">E85</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="newton.html#K13">SetPrimes</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E22:51_3_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">F1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E10:51_3_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_1.html#T42">FINSEQ_1:42</a></i>;<br/>
<a NAME="E23:51_3_1"/><b>then </b><i><font color="Green">E86</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">F1</font> <a href="tarski.html#R1">c=</a>  <a href="newton.html#K13">SetPrimes</a> 
 
<b>by </b><i><a class="txt" href="nat_4.html#E15:51_3_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">F1</font> = <font color="Maroon">F1</font> as    <a href="trees_4.html#M1">FinSequence</a> of  <a href="newton.html#K13">SetPrimes</a>  <b>by </b><i>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E25:51_3_1"/><i><font color="Green">E87</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="nat_4.html#E12:51_3_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<i><font color="Green">E88</font></i>:  <a href="wsierp_1.html#K8">Product</a> <font color="Maroon">F</font> = 
 <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="nat_4.html#E3:51"><i><font color="Green">E54</font></i></a>, <a class="txt" href="nat_4.html#E5:51_3_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="nat_4.html#E9:51_3_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="nat_4.html#E8:51_3_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="finseq_3.html#T48">FINSEQ_3:48</a></i>
<br/>.= 
 <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_3.html#T50">FINSEQ_3:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <font color="Maroon">F1</font></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T126">RVSUM_1:126</a></i>
;<br/>
<div><i><font color="Green">E90</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:51_3_1_5"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F1</font>
 ;<br/><a NAME="E2:51_3_1_5"/><b>then </b>
<font color="Maroon">F1</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F1</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E3:51_3_1_5"/><b>then </b>
<font color="Maroon">F1</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font> is   <a href="newton.html#NM1">Prime</a>
 <b>by </b><i>, <a class="ref" href="newton.html#D6">NEWTON:def 6</a></i>;<br/><b>hence </b><a NAME="E4:51_3_1_5"/>
<font color="Maroon">F1</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/></div><b>end;</b></div>
<a NAME="E28:51_3_1"/><i><font color="Green">E91</font></i>: 
 <a href="wsierp_1.html#K8">Product</a> <font color="Maroon">F1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E29:51_3_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <font color="Maroon">F1</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="nat_4.html#E3:51"><i><font color="Green">E54</font></i></a>, , <a class="ref" href="nat_3.html#T28">NAT_3:28</a></i>;<br/>
<a NAME="E30:51_3_1"/><i><font color="Green">E93</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="xboole_0.html#K4">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="nat_4.html#E5:51_3_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/>
<div><i><font color="Green">E94</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3_1_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:51_3_1_6"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font>
 ;<br/><a NAME="E2:51_3_1_6"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E10:51_3_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/><a NAME="E3:51_3_1_6"/><b>then </b><i><font color="Green">E95</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F1</font> or <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">x</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:51_3_1_6"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:51_3_1_6_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F1</font> or <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 </span><b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_6_1_1"><b>suppose </b></a><a NAME="E1:51_3_1_6_1_1"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F1</font>
 ;<br/><div class="add"><a NAME="E2:51_3_1_6_1_1"/>
not <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">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="nat_4.html#E17:51_3_1"><i><font color="Green">E81</font></i></a>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:51_3_1_6_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:51_3_1_6_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_3.html#T24">NAT_3:24</a>, <a class="txt" href="nat_4.html#E16:51_3_1"><i><font color="Green">E80</font></i></a></i>;<br/><b>hence </b><a NAME="E5:51_3_1_6_1_1"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="nat_4.html#E17:51_3_1"><i><font color="Green">E81</font></i></a>, , , <a class="txt" href="nat_4.html#E3:51_3_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="nat_4.html#E5:51_3_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="nat_4.html#E14:51_3_1"><i><font color="Green">E69</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_6_1_2"><b>suppose </b></a><a NAME="E1:51_3_1_6_1_2"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:51_3_1_6_1_2"/>
<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">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="nat_4.html#E17:51_3_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:51_3_1_6_1_2"/><b>then </b>
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F1</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:51_3_1_6_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <font color="Maroon">F1</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="txt" href="nat_4.html#E3:51_3_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="nat_4.html#E5:51_3_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="nat_4.html#E14:51_3_1"><i><font color="Green">E69</font></i></a></i>;<br/><b>hence </b><a NAME="E5:51_3_1_6_1_2"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="nat_4.html#E17:51_3_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="nat_4.html#E16:51_3_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="nat_3.html#T22">NAT_3:22</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3_1_7"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:51_3_1_7"/>
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font>
 ;<br/><a NAME="E2:51_3_1_7"/><b>then </b>
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E10:51_3_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/><a NAME="E3:51_3_1_7"/><b>then </b>
( not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F1</font> &amp; not <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">x</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:51_3_1_7"/><b>then </b><i><font color="Green">E100</font></i>: 
( <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <font color="Maroon">F1</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="txt" href="nat_4.html#E3:51_3_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="nat_4.html#E5:51_3_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="nat_4.html#E14:51_3_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:51_3_1_7"/>
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/><b>hence </b><a NAME="E6:51_3_1_7"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , <a class="ref" href="nat_3.html#T24">NAT_3:24</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E33:51_3_1"/>
( ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> implies <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1 ) &amp; ( not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> implies <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:51_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:51"/><i><font color="Green">E101</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><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">p</font> be   <a href="newton.html#NM1">Prime</a>;<br/>







<b>reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i/>;<br/>
<a NAME="E6:51"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 
;<br/>
<b>hence </b><a NAME="E7:51"/>
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="newton.html#K13">SetPrimes</a>  &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font> &amp; <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K8">Product</a> <font color="Maroon">F</font> implies ( ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> implies <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1 ) &amp; ( not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> implies <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 ) ) )
 <b>by </b><i/>;<br/>


</div>
