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<div class="add">

<b>let </b><font color="Maroon" title="c1">p</font> be   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>;<br/><b>let </b><font color="Maroon" title="c2">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c3">m</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c4">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>







<b>assume </b><a NAME="E1:68_1"/><i><font color="Green" title="E64">A1</font></i>: 
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> 
 ;<br/>

<b>assume </b><a NAME="E2:68_1"/><i><font color="Green" title="E65">A2</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c5">XX</font> =  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</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" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:68_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ;<br/><a NAME="E2:68_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">y'</font> being <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">y'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">y'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">y'</font>] )
 <b>by </b><i><a class="txt" href="nat_4.html#E1:68_1"><i><font color="Green" title="E64">A1</font></i></a></i>;<br/><a NAME="E3:68_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">b</font> &amp; <font color="Olive" title="b1">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> )
 <b>by </b><i><a class="ref" href="nat_4.html#T16" target="_self" title="NAT_4:th.16">Th16</a></i>;<br/><b>hence </b><a NAME="E4:68_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:68_1"/><b>then </b><i><font color="Green" title="E66">A3</font></i>: 
<font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E5:68_1"/>
for <font color="Olive" title="b1">p</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a><br/> for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b3">m</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b4">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b5">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b5">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Olive" title="b3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b5">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">n</font>,<font color="Olive" title="b3">m</font>,<font color="Olive" title="b5">p'</font>] </span>}</span>  holds <br/> <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Olive" title="b4">X</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>from </b><i><a class="ref" href="nat_4.html#S1" target="_self" title="NAT_4:sch.1">NAT_4:sch 1</a>();<br/></i>
<a NAME="E6:68_1"/><b>then </b><i><font color="Green" title="E67">A4</font></i>: 
 <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:68_1"><i><font color="Green" title="E64">A1</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>] <b>means </b>( <font color="Maroon">P<sub>1</sub></font>[$1,$2,$3] &amp; $3 <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default">$3 <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> $2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> $2</span>)</span> );<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>] <b>means </b>( <font color="Maroon">P<sub>1</sub></font>[$1,$2,$3] &amp; $3 <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default">$3 <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> $2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> $2</span>)</span> );<br/>
<b>set </b><font color="Maroon" title="c6">X1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c7">X2</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</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" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:68_1_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> 
 ;<br/><a NAME="E2:68_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">p'</font> being <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] )
 ;<br/><b>hence </b><a NAME="E3:68_1_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:68_1"><i><font color="Green" title="E64">A1</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:68_1"/><b>then </b><i><font color="Green" title="E68">A5</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">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" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:68_1_3"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> 
 ;<br/><a NAME="E2:68_1_3"/><b>then </b>
 ex <font color="Olive" title="b1">p'</font> being <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] )
 ;<br/><b>hence </b><a NAME="E3:68_1_3"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:68_1"><i><font color="Green" title="E64">A1</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:68_1"/><b>then </b><i><font color="Green" title="E69">A6</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E11:68_1"/><i><font color="Green" title="E70">A7</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E4:68_1"><i><font color="Green" title="E66">A3</font></i></a>, <a class="txt" href="nat_4.html#E8:68_1"><i><font color="Green" title="E68">A5</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E12:68_1"/><i><font color="Green" title="E71">A8</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E4:68_1"><i><font color="Green" title="E66">A3</font></i></a>, <a class="txt" href="nat_4.html#E10:68_1"><i><font color="Green" title="E69">A6</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<div><i><font color="Green" title="E72">A9</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" title="c8">k1</font>, <font color="Maroon" title="c9">k2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:68_1_4"/><i><font color="Green" title="E73">A10</font></i>: 
( <font color="Maroon" title="c8">k1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span>  &amp; <font color="Maroon" title="c9">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span>  )
 ;<br/><a NAME="E2:68_1_4"/><b>then </b><i><font color="Green" title="E74">A11</font></i>: 
 ex <font color="Olive" title="b1">p1</font> being <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k1</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p1</font>] )
 ;<br/><a NAME="E3:68_1_4"/>
 ex <font color="Olive" title="b1">p2</font> being <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">p2</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p2</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k2</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p2</font>] )
 <b>by </b><i><a class="txt" href="nat_4.html#E1:68_1_4"><i><font color="Green" title="E73">A10</font></i></a></i>;<br/><b>hence </b><a NAME="E4:68_1_4"/>
<font color="Maroon" title="c8">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c9">k2</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E2:68_1_4"><i><font color="Green" title="E74">A11</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:68_1"/><i><font color="Green" title="E73">A12</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">X</font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E8:68_1"><i><font color="Green" title="E68">A5</font></i></a>, <a class="txt" href="nat_4.html#E10:68_1"><i><font color="Green" title="E69">A6</font></i></a>, <a class="ref" href="xboole_1.html#T13" title="XBOOLE_1:th.13">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" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:68_1_5"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c9">p'</font> being  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:68_1_5"/><i><font color="Green" title="E74">A13</font></i>: 
( <font color="Maroon" title="c9">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c9">p'</font>] )
 <b>by </b><i><a class="txt" href="nat_4.html#E1:68_1"><i><font color="Green" title="E64">A1</font></i></a></i>;<br/><a NAME="E4:68_1_5"/>
( ( <font color="Maroon" title="c9">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c9">p'</font>] ) or ( <font color="Maroon" title="c9">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c9">p'</font>] ) )
 <b>by </b><i><a class="txt" href="nat_4.html#E3:68_1_5"><i><font color="Green" title="E74">A13</font></i></a></i>;<br/><a NAME="E5:68_1_5"/><b>then </b>
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span>  or <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span>  )
 ;<br/><b>hence </b><a NAME="E6:68_1_5"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E16:68_1"/><b>then </b><i><font color="Green" title="E74">A14</font></i>: 
<font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>then </b><i><font color="Green" title="E75">A15</font></i>:  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> = 
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</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" title="E73">A12</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>
<br/>.= 
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> </span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E11:68_1"><i><font color="Green" title="E70">A7</font></i></a>, <a class="txt" href="nat_4.html#E12:68_1"><i><font color="Green" title="E71">A8</font></i></a>, <a class="txt" href="nat_4.html#E13:68_1"><i><font color="Green" title="E72">A9</font></i></a>, <a class="ref" href="finseq_3.html#T48" title="FINSEQ_3:th.48">FINSEQ_3:48</a></i>
<br/>.= 
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> </span>)</span></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="nat_4.html#T51" target="_self" title="NAT_4:th.51">Th51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> </span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T105" title="RVSUM_1:th.105">RVSUM_1:105</a></i>
;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>] <b>means </b>( <font color="Maroon">P<sub>1</sub></font>[$1,$2,$3] &amp; $3 <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default">$3 <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> $2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> $2</span>)</span> );<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>] <b>means </b>( <font color="Maroon">P<sub>1</sub></font>[$1,$2,$3] &amp; $3 <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default">$3 <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> $2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> $2</span>)</span> );<br/>
<b>set </b><font color="Maroon" title="c8">X11</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c9">X12</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</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" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:68_1_7"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> 
 ;<br/><a NAME="E2:68_1_7"/><b>then </b>
 ex <font color="Olive" title="b1">p'</font> being <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] )
 ;<br/><b>hence </b><a NAME="E3:68_1_7"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> 
 ;<br/></div><b>end;</b></div>
<a NAME="E19:68_1"/><b>then </b><i><font color="Green" title="E76">A16</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">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" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:68_1_8"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> 
 ;<br/><a NAME="E2:68_1_8"/><b>then </b>
 ex <font color="Olive" title="b1">p'</font> being <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] )
 ;<br/><b>hence </b><a NAME="E3:68_1_8"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> 
 ;<br/></div><b>end;</b></div>
<a NAME="E21:68_1"/><b>then </b><i><font color="Green" title="E77">A17</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E22:68_1"/><i><font color="Green" title="E78">A18</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E11:68_1"><i><font color="Green" title="E70">A7</font></i></a>, <a class="txt" href="nat_4.html#E19:68_1"><i><font color="Green" title="E76">A16</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E23:68_1"/><i><font color="Green" title="E79">A19</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E11:68_1"><i><font color="Green" title="E70">A7</font></i></a>, <a class="txt" href="nat_4.html#E21:68_1"><i><font color="Green" title="E77">A17</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<div><i><font color="Green" title="E80">A20</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" title="c10">k1</font>, <font color="Maroon" title="c11">k2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:68_1_9"/><i><font color="Green" title="E81">A21</font></i>: 
( <font color="Maroon" title="c10">k1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span>  &amp; <font color="Maroon" title="c11">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span>  )
 ;<br/><a NAME="E2:68_1_9"/><b>then </b><i><font color="Green" title="E82">A22</font></i>: 
 ex <font color="Olive" title="b1">p1</font> being <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k1</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p1</font>] )
 ;<br/><a NAME="E3:68_1_9"/>
 ex <font color="Olive" title="b1">p2</font> being <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">p2</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p2</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k2</font> &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p2</font>] )
 <b>by </b><i><a class="txt" href="nat_4.html#E1:68_1_9"><i><font color="Green" title="E81">A21</font></i></a></i>;<br/><b>hence </b><a NAME="E4:68_1_9"/>
<font color="Maroon" title="c10">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c11">k2</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E2:68_1_9"><i><font color="Green" title="E82">A22</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E25:68_1"/><i><font color="Green" title="E81">A23</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="nat_4.html#E19:68_1"><i><font color="Green" title="E76">A16</font></i></a>, <a class="txt" href="nat_4.html#E21:68_1"><i><font color="Green" title="E77">A17</font></i></a>, <a class="ref" href="xboole_1.html#T13" title="XBOOLE_1:th.13">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" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:68_1_10"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c11">p'</font> being  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:68_1_10"/><i><font color="Green" title="E82">A24</font></i>: 
( <font color="Maroon" title="c11">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c11">p'</font>] )
 ;<br/><a NAME="E4:68_1_10"/>
( ( <font color="Maroon" title="c11">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c11">p'</font>] ) or ( <font color="Maroon" title="c11">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c11">p'</font>] ) )
 <b>by </b><i><a class="txt" href="nat_4.html#E3:68_1_10"><i><font color="Green" title="E82">A24</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E5:68_1_10"/><b>then </b>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span>  or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span>  )
 ;<br/><b>hence </b><a NAME="E6:68_1_10"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E27:68_1"/><b>then </b><i><font color="Green" title="E82">A25</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>then </b><i><font color="Green" title="E83">A26</font></i>:  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p5">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> </span>)</span></span>)</span> = 
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</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" title="E81">A23</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>
<br/>.= 
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> </span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E22:68_1"><i><font color="Green" title="E78">A18</font></i></a>, <a class="txt" href="nat_4.html#E23:68_1"><i><font color="Green" title="E79">A19</font></i></a>, <a class="txt" href="nat_4.html#E24:68_1"><i><font color="Green" title="E80">A20</font></i></a>, <a class="ref" href="finseq_3.html#T48" title="FINSEQ_3:th.48">FINSEQ_3:48</a></i>
<br/>.= 
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> </span>)</span></span>)</span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="nat_4.html#T51" target="_self" title="NAT_4:th.51">Th51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> </span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T105" title="RVSUM_1:th.105">RVSUM_1:105</a></i>
;<br/>
<div><i><font color="Green" title="E84">A27</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" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> 
 ;<br/><a NAME="E2:68_1_12"/><b>then </b>
 ex <font color="Olive" title="b1">p'</font> being <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</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" title="b1">p</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a><br/> for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b3">m</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b4">X11</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">X11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b5">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b5">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Olive" title="b3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b5">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b2">n</font>,<font color="Olive" title="b3">m</font>,<font color="Olive" title="b5">p'</font>] </span>}</span>  holds <br/> <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Olive" title="b4">X11</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>from </b><i><a class="ref" href="nat_4.html#S1" target="_self" title="NAT_4:sch.1">NAT_4:sch 1</a>();<br/></i>
<a NAME="E31:68_1"/><b>then </b><i><font color="Green" title="E85">A28</font></i>: 
 <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> </span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<a NAME="E32:68_1"/>
for <font color="Olive" title="b1">p</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a><br/> for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b3">m</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b4">X11</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">X11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b5">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b5">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Olive" title="b3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b5">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b2">n</font>,<font color="Olive" title="b3">m</font>,<font color="Olive" title="b5">p'</font>] </span>}</span>  &amp; not <font color="Olive" title="b1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">X11</font> holds <br/><font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Olive" title="b4">X11</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>from </b><i><a class="ref" href="nat_4.html#S2" target="_self" title="NAT_4:sch.2">NAT_4:sch 2</a>();<br/></i>
<a NAME="E33:68_1"/><b>then </b><i><font color="Green" title="E86">A29</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p5">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="nat_4.html#E29:68_1"><i><font color="Green" title="E84">A27</font></i></a></i>;<br/>
<div><i><font color="Green" title="E87">A30</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" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> 
 ;<br/><a NAME="E2:68_1_13"/><b>then </b>
 ex <font color="Olive" title="b1">p'</font> being <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</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" title="b1">p</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a><br/> for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b3">m</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b4">X2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b5">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b5">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Olive" title="b3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b5">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">n</font>,<font color="Olive" title="b3">m</font>,<font color="Olive" title="b5">p'</font>] </span>}</span>  holds <br/> <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Olive" title="b4">X2</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>from </b><i><a class="ref" href="nat_4.html#S1" target="_self" title="NAT_4:sch.1">NAT_4:sch 1</a>();<br/></i>
<a NAME="E36:68_1"/><b>then </b><i><font color="Green" title="E88">A31</font></i>: 
 <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> </span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<a NAME="E37:68_1"/>
for <font color="Olive" title="b1">p</font> being  <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a><br/> for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b3">m</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b4">X2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b5">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b5">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Olive" title="b3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b5">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">n</font>,<font color="Olive" title="b3">m</font>,<font color="Olive" title="b5">p'</font>] </span>}</span>  &amp; not <font color="Olive" title="b1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">X2</font> holds <br/><font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Olive" title="b4">X2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>from </b><i><a class="ref" href="nat_4.html#S2" target="_self" title="NAT_4:sch.2">NAT_4:sch 2</a>();<br/></i>
<a NAME="E38:68_1"/><b>then </b><i><font color="Green" title="E89">A32</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p5">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="nat_4.html#E34:68_1"><i><font color="Green" title="E87">A30</font></i></a></i>;<br/>
<a NAME="E39:68_1"/>
<font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="nat_4.html#E2:68_1"><i><font color="Green" title="E65">A2</font></i></a>, <a class="txt" href="nat_4.html#E16:68_1"><i><font color="Green" title="E74">A14</font></i></a>, <a class="txt" href="nat_4.html#E34:68_1"><i><font color="Green" title="E87">A30</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E40:68_1"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="nat_4.html#E27:68_1"><i><font color="Green" title="E82">A25</font></i></a>, <a class="txt" href="nat_4.html#E29:68_1"><i><font color="Green" title="E84">A27</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E41:68_1"/><b>then </b>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b2">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b2">p'</font>] </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E42:68_1"/><b>then </b><i><font color="Green" title="E90">A33</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">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" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:68_1_14"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> 
 ;<br/><a NAME="E2:68_1_14"/><b>then </b>
 ex <font color="Olive" title="b1">p'</font> being <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] )
 ;<br/><b>hence </b><a NAME="E3:68_1_14"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E44:68_1"/><b>then </b><i><font color="Green" title="E91">A34</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">m1</font> = <font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">m1</font> = <font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E46:68_1"/><i><font color="Green" title="E92">A35</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>
<i><font color="Green" title="E93">A36</font></i>: <font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">p</font></span>)</span>
<b>by </b><i><a class="ref" href="nat_3.html#T32" title="NAT_3:th.32">NAT_3:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 1
<b>by </b><i><a class="txt" href="nat_4.html#E46:68_1"><i><font color="Green" title="E92">A35</font></i></a>, <a class="ref" href="nat_3.html#T22" title="NAT_3:th.22">NAT_3:22</a></i>
;<br/>
<font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> </span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</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" title="E67">A4</font></i></a>, <a class="txt" href="nat_4.html#E17:68_1"><i><font color="Green" title="E75">A15</font></i></a>, <a class="ref" href="nat_4.html#T53" target="_self" title="NAT_4:th.53">Th53</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</font>] </span>}</span> </span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="nat_4.html#E36:68_1"><i><font color="Green" title="E88">A31</font></i></a>, <a class="txt" href="nat_4.html#E38:68_1"><i><font color="Green" title="E89">A32</font></i></a>, <a class="ref" href="nat_4.html#T53" target="_self" title="NAT_4:th.53">Th53</a></i>
<br/>.= 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p'</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p'</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p'</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" title="E83">A26</font></i></a>, <a class="txt" href="nat_4.html#E31:68_1"><i><font color="Green" title="E85">A28</font></i></a>, <a class="txt" href="nat_4.html#E33:68_1"><i><font color="Green" title="E86">A29</font></i></a>, <a class="ref" href="nat_4.html#T53" target="_self" title="NAT_4:th.53">Th53</a></i>
<br/>.= 
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E42:68_1"><i><font color="Green" title="E90">A33</font></i></a>, <a class="txt" href="nat_4.html#E44:68_1"><i><font color="Green" title="E91">A34</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>
<br/>.= 
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c11">m1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_3.html#T50" title="FINSEQ_3:th.50">FINSEQ_3:50</a></i>
<br/>.= 
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c11">m1</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
<b>by </b><i><a class="ref" href="nat_4.html#T52" target="_self" title="NAT_4:th.52">Th52</a></i>
<br/>.= 
<font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c11">m1</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T103" title="RVSUM_1:th.103">RVSUM_1:103</a></i>
;<br/>
<b>hence </b><a NAME="E49:68_1"/>
<font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E47:68_1"><i><font color="Green" title="E93">A36</font></i></a></i>;<br/>


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