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<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_3.html#K9" title="NAT_3:func.9">|^</a> <font color="Maroon" title="c2">m</font></span><br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="newton.html#K13" title="NEWTON:func.13">SetPrimes</a>  implies <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_3.html#K9" title="NAT_3:func.9">|^</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><b>assume </b><a NAME="E1:100_1"/>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="newton.html#K13" title="NEWTON:func.13">SetPrimes</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_3.html#K9" title="NAT_3:func.9">|^</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><b>then reconsider </b><font color="Maroon" title="c4">p</font> = <font color="Maroon" title="c3">i</font> as  <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>by </b><i><a class="ref" href="newton.html#D6" title="NEWTON:def.6">NEWTON:def 6</a></i>;<br/><a NAME="E3:100_1"/><i><font color="Green" title="E53">A1</font></i>: 
<font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:100_1"/><i><font color="Green" title="E54">A2</font></i>: 
<font color="Maroon" title="c4">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">n</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_3.html#T32" target="_self" title="NAT_3:th.32">Th32</a></i>;<br/><a NAME="E5:100_1"/><i><font color="Green" title="E55">A3</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_3.html#K9" title="NAT_3:func.9">|^</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">m</font>
 <b>by </b><i><a class="ref" href="nat_3.html#D6" target="_self" title="NAT_3:def.6">Def6</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:100_1_1"/>
( <font color="Maroon" title="c4">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c4">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_1_1_1"><b>suppose </b></a><a NAME="E1:100_1_1_1"/><i><font color="Green" title="E56">A4</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_3.html#K9" title="NAT_3:func.9">|^</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="#E4:100_1"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="nat_3.html#T55" target="_self" title="NAT_3:th.55">Th55</a></i>
<br/>.= 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">m</font>
<b>by </b><i><a class="txt" href="#E3:100_1"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="newton.html#T16" title="NEWTON:th.16">NEWTON:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_3.html#K9" title="NAT_3:func.9">|^</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
<b>by </b><i><a class="txt" href="#E5:100_1"><i><font color="Green" title="E55">A3</font></i></a>, <a class="txt" href="#E1:100_1_1_1"><i><font color="Green" title="E56">A4</font></i></a>, <a class="ref" href="nat_3.html#T55" target="_self" title="NAT_3:th.55">Th55</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_1_1_2"><b>suppose </b></a><a NAME="E1:100_1_1_2"/><i><font color="Green" title="E56">A5</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_3.html#K9" title="NAT_3:func.9">|^</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font> = 
<font color="Maroon" title="c4">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E4:100_1"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="nat_3.html#T56" target="_self" title="NAT_3:th.56">Th56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">m</font>
<b>by </b><i><a class="txt" href="#E4:100_1"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="newton.html#T14" title="NEWTON:th.14">NEWTON:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_3.html#K9" title="NAT_3:func.9">|^</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
<b>by </b><i><a class="txt" href="#E5:100_1"><i><font color="Green" title="E55">A3</font></i></a>, <a class="txt" href="#E1:100_1_1_2"><i><font color="Green" title="E56">A5</font></i></a>, <a class="ref" href="nat_3.html#T56" target="_self" title="NAT_3:th.56">Th56</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:100"/>
 <a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_3.html#K9" title="NAT_3:func.9">|^</a> <font color="Maroon" title="c2">m</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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