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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="newton.html#NM1">Prime</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>







<b>assume </b><a NAME="E1:68_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 ;<br/>

<b>assume </b><a NAME="E2:68_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:68_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:68_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="newton.html#NM1">Prime</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] )
 <b>by </b><i><a class="txt" href="nat_4.html#E1:68_1"><i><font color="Green">E64</font></i></a></i>;<br/><a NAME="E3:68_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="nat_4.html#T16" target="_self">Th16</a></i>;<br/><b>hence </b><a NAME="E4:68_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:68_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E5:68_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="newton.html#NM1">Prime</a><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>3</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>4</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>5</sub></font>] </span>}</span>  holds <br/> <a href="wsierp_1.html#K9">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>from </b><i><a class="ref" href="nat_4.html#S1" target="_self">NAT_4:sch 1</a>();<br/></i>
<a NAME="E6:68_1"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="wsierp_1.html#K9">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:68_1"><i><font color="Green">E64</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,  <a href="nat_1.html#NM1">Nat</a>,  <a href="newton.html#NM1">Prime</a>] means ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>3</sub></font>] &amp; <font color="Maroon">a<sub>3</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">a<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">a<sub>2</sub></font></span>)</span> );<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,  <a href="nat_1.html#NM1">Nat</a>,  <a href="newton.html#NM1">Prime</a>] means ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>3</sub></font>] &amp; <font color="Maroon">a<sub>3</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">a<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">a<sub>2</sub></font></span>)</span> );<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:68_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 ;<br/><a NAME="E2:68_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="newton.html#NM1">Prime</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] )
 ;<br/><b>hence </b><a NAME="E3:68_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:68_1"><i><font color="Green">E64</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:68_1"/><b>then </b><i><font color="Green">E68</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:68_1_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 ;<br/><a NAME="E2:68_1_3"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="newton.html#NM1">Prime</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] )
 ;<br/><b>hence </b><a NAME="E3:68_1_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:68_1"><i><font color="Green">E64</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:68_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E11:68_1"/><i><font color="Green">E70</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E8:68_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="nat_4.html#E4:68_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E12:68_1"/><i><font color="Green">E71</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E10:68_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="nat_4.html#E4:68_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<div><i><font color="Green">E72</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:68_1_4"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span>  &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span>  )
 ;<br/><a NAME="E2:68_1_4"/><b>then </b><i><font color="Green">E74</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="newton.html#NM1">Prime</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] )
 ;<br/><a NAME="E3:68_1_4"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="newton.html#NM1">Prime</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] )
 <b>by </b><i><a class="txt" href="nat_4.html#E1:68_1_4"><i><font color="Green">E73</font></i></a></i>;<br/><b>hence </b><a NAME="E4:68_1_4"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, <a class="txt" href="nat_4.html#E2:68_1_4"><i><font color="Green">E74</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:68_1"/><i><font color="Green">E73</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E8:68_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="nat_4.html#E10:68_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:68_1_5"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="newton.html#NM1">Prime</a><b> such that </b><br/><a NAME="E3:68_1_5"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font>] )
 <b>by </b><i><a class="txt" href="nat_4.html#E1:68_1_5"><i><font color="Green">E74</font></i></a>, <a class="txt" href="nat_4.html#E1:68_1"><i><font color="Green">E64</font></i></a></i>;<br/><a NAME="E4:68_1_5"/>
( ( <font color="Maroon">c<sub>9</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font>] ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>9</sub></font>] ) )
 <b>by </b><i><a class="txt" href="nat_4.html#E3:68_1_5"><i><font color="Green">E75</font></i></a></i>;<br/><a NAME="E5:68_1_5"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span>  or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span>  )
 ;<br/><b>hence </b><a NAME="E6:68_1_5"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E16:68_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<i><font color="Green">E75</font></i>:  <a href="wsierp_1.html#K8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> = 
 <a href="wsierp_1.html#K8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p0">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p0">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E14:68_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="nat_4.html#E16:68_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>
<br/>.= 
 <a href="wsierp_1.html#K8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p0">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> </span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p0">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_3.html#T48">FINSEQ_3:48</a>, <a class="txt" href="nat_4.html#E13:68_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="nat_4.html#E11:68_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="nat_4.html#E12:68_1"><i><font color="Green">E71</font></i></a></i>
<br/>.= 
 <a href="wsierp_1.html#K8">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p0">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> </span>)</span></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p0">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="nat_4.html#T51" target="_self">Th51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p0">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> </span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p0">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T105">RVSUM_1:105</a></i>
;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,  <a href="nat_1.html#NM1">Nat</a>,  <a href="newton.html#NM1">Prime</a>] means ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>3</sub></font>] &amp; <font color="Maroon">a<sub>3</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">a<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">a<sub>2</sub></font></span>)</span> );<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,  <a href="nat_1.html#NM1">Nat</a>,  <a href="newton.html#NM1">Prime</a>] means ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>3</sub></font>] &amp; <font color="Maroon">a<sub>3</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">a<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">a<sub>2</sub></font></span>)</span> );<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:68_1_7"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 ;<br/><a NAME="E2:68_1_7"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="newton.html#NM1">Prime</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] )
 ;<br/><b>hence </b><a NAME="E3:68_1_7"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 ;<br/></div><b>end;</b></div>
<a NAME="E19:68_1"/><b>then </b><i><font color="Green">E76</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:68_1_8"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 ;<br/><a NAME="E2:68_1_8"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="newton.html#NM1">Prime</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] )
 ;<br/><b>hence </b><a NAME="E3:68_1_8"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 ;<br/></div><b>end;</b></div>
<a NAME="E21:68_1"/><b>then </b><i><font color="Green">E77</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E22:68_1"/><i><font color="Green">E78</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E19:68_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="nat_4.html#E11:68_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E23:68_1"/><i><font color="Green">E79</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E21:68_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="nat_4.html#E11:68_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<div><i><font color="Green">E80</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:68_1_9"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span>  &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span>  )
 ;<br/><a NAME="E2:68_1_9"/><b>then </b><i><font color="Green">E82</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="newton.html#NM1">Prime</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] )
 ;<br/><a NAME="E3:68_1_9"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="newton.html#NM1">Prime</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] )
 <b>by </b><i><a class="txt" href="nat_4.html#E1:68_1_9"><i><font color="Green">E81</font></i></a></i>;<br/><b>hence </b><a NAME="E4:68_1_9"/>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="nat_4.html#E2:68_1_9"><i><font color="Green">E82</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E25:68_1"/><i><font color="Green">E81</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="nat_4.html#E19:68_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="nat_4.html#E21:68_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_10"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:68_1_10"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 ;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="newton.html#NM1">Prime</a><b> such that </b><br/><a NAME="E3:68_1_10"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>11</sub></font>] )
 <b>by </b><i><a class="txt" href="nat_4.html#E1:68_1_10"><i><font color="Green">E82</font></i></a></i>;<br/><a NAME="E4:68_1_10"/>
( ( <font color="Maroon">c<sub>11</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>11</sub></font>] ) or ( <font color="Maroon">c<sub>11</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>11</sub></font>] ) )
 <b>by </b><i><a class="txt" href="nat_4.html#E3:68_1_10"><i><font color="Green">E83</font></i></a>, <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/><a NAME="E5:68_1_10"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span>  or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span>  )
 ;<br/><b>hence </b><a NAME="E6:68_1_10"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E27:68_1"/><b>then </b><i><font color="Green">E82</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<i><font color="Green">E83</font></i>:  <a href="wsierp_1.html#K8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> </span>)</span></span>)</span> = 
 <a href="wsierp_1.html#K8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p0">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p0">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E25:68_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="nat_4.html#E27:68_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>
<br/>.= 
 <a href="wsierp_1.html#K8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p0">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> </span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p0">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_3.html#T48">FINSEQ_3:48</a>, <a class="txt" href="nat_4.html#E24:68_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="nat_4.html#E22:68_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="nat_4.html#E23:68_1"><i><font color="Green">E79</font></i></a></i>
<br/>.= 
 <a href="wsierp_1.html#K8">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p0">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> </span>)</span></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p0">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="nat_4.html#T51" target="_self">Th51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p0">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> </span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p0">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T105">RVSUM_1:105</a></i>
;<br/>
<div><i><font color="Green">E84</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_12"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:68_1_12"/>
<font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 ;<br/><a NAME="E2:68_1_12"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="newton.html#NM1">Prime</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] )
 ;<br/><b>hence </b><a NAME="E3:68_1_12"/>
contradiction
 ;<br/></div><b>end;</b></div>
<a NAME="E30:68_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="newton.html#NM1">Prime</a><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>3</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>4</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>5</sub></font>] </span>}</span>  holds <br/> <a href="wsierp_1.html#K9">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>from </b><i><a class="ref" href="nat_4.html#S1" target="_self">NAT_4:sch 1</a>();<br/></i>
<a NAME="E31:68_1"/><b>then </b><i><font color="Green">E85</font></i>: 
 <a href="wsierp_1.html#K9">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
;<br/>
<a NAME="E32:68_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="newton.html#NM1">Prime</a><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>3</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>4</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>5</sub></font>] </span>}</span>  &amp; not <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K9">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Olive">b<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>from </b><i><a class="ref" href="nat_4.html#S2" target="_self">NAT_4:sch 2</a>();<br/></i>
<a NAME="E33:68_1"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K9">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="nat_4.html#E29:68_1"><i><font color="Green">E84</font></i></a></i>;<br/>
<div><i><font color="Green">E87</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_13"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:68_1_13"/>
<font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 ;<br/><a NAME="E2:68_1_13"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="newton.html#NM1">Prime</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] )
 ;<br/><b>hence </b><a NAME="E3:68_1_13"/>
contradiction
 ;<br/></div><b>end;</b></div>
<a NAME="E35:68_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="newton.html#NM1">Prime</a><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>3</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>4</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>5</sub></font>] </span>}</span>  holds <br/> <a href="wsierp_1.html#K9">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>from </b><i><a class="ref" href="nat_4.html#S1" target="_self">NAT_4:sch 1</a>();<br/></i>
<a NAME="E36:68_1"/><b>then </b><i><font color="Green">E88</font></i>: 
 <a href="wsierp_1.html#K9">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
;<br/>
<a NAME="E37:68_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="newton.html#NM1">Prime</a><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>3</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>4</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>5</sub></font>] </span>}</span>  &amp; not <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K9">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Olive">b<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>from </b><i><a class="ref" href="nat_4.html#S2" target="_self">NAT_4:sch 2</a>();<br/></i>
<a NAME="E38:68_1"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K9">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="nat_4.html#E34:68_1"><i><font color="Green">E87</font></i></a></i>;<br/>
<a NAME="E39:68_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="nat_4.html#E2:68_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="nat_4.html#E16:68_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="nat_4.html#E34:68_1"><i><font color="Green">E87</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E40:68_1"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="nat_4.html#E27:68_1"><i><font color="Green">E82</font></i></a>, <a class="txt" href="nat_4.html#E29:68_1"><i><font color="Green">E84</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E41:68_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="nat_4.html#E40:68_1"><i><font color="Green">E90</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E42:68_1"/><b>then </b><i><font color="Green">E91</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_14"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:68_1_14"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 ;<br/><a NAME="E2:68_1_14"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="newton.html#NM1">Prime</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] )
 ;<br/><b>hence </b><a NAME="E3:68_1_14"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E44:68_1"/><b>then </b><i><font color="Green">E92</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E46:68_1"/><i><font color="Green">E93</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<i><font color="Green">E94</font></i>: <font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="nat_3.html#T32">NAT_3:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K24">*</a> 1
<b>by </b><i><a class="ref" href="nat_3.html#T22">NAT_3:22</a>, <a class="txt" href="nat_4.html#E46:68_1"><i><font color="Green">E93</font></i></a></i>
;<br/>
<font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K9">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p0">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> </span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p0">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E6:68_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="nat_4.html#T53" target="_self">Th53</a>, <a class="txt" href="nat_4.html#E17:68_1"><i><font color="Green">E75</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p0">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> </span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> 0
<b>by </b><i><a class="txt" href="nat_4.html#E38:68_1"><i><font color="Green">E89</font></i></a>, <a class="txt" href="nat_4.html#E36:68_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="nat_4.html#T53" target="_self">Th53</a></i>
<br/>.= 
0 <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p0">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] </span>}</span> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E28:68_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="nat_4.html#E31:68_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="nat_4.html#E33:68_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="nat_4.html#T53" target="_self">Th53</a></i>
<br/>.= 
 <a href="wsierp_1.html#K8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_4.html#K1" target="_self">|-count</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"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E44:68_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="nat_4.html#E42:68_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>
<br/>.= 
 <a href="wsierp_1.html#K8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>11</sub></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/>.= 
 <a href="wsierp_1.html#K8">Sum</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i><a class="ref" href="nat_4.html#T52" target="_self">Th52</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T103">RVSUM_1:103</a></i>
;<br/>
<b>hence </b><a NAME="E49:68_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K9">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="nat_4.html#E47:68_1"><i><font color="Green">E94</font></i></a></i>;<br/>


</div>
