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<b>assume </b><a NAME="E1:99"/><i><font color="Green" title="E55">A1</font></i>: 
 <a href="newton.html#K13" title="NEWTON:func.13">SetPrimes</a>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c1">SP</font> =  <a href="newton.html#K13" title="NEWTON:func.13">SetPrimes</a>  as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>consider </b><font color="Maroon" title="c2">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:99"/><i><font color="Green" title="E56">A2</font></i>: 
 <a href="newton.html#K13" title="NEWTON:func.13">SetPrimes</a> , <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 <b>by </b><i><a class="txt" href="newton.html#E1:99"><i><font color="Green" title="E55">A1</font></i></a>, <a class="ref" href="finseq_1.html#T77" title="FINSEQ_1:th.77">FINSEQ_1:77</a></i>;<br/>
<b>set </b><font color="Maroon" title="c3">p</font> =  <a href="newton.html#K15" title="NEWTON:func.15">primenumber</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/>
<a NAME="E5:99"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <a href="newton.html#K13" title="NEWTON:func.13">SetPrimes</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="newton.html#E4:99"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></i>;<br/>
<a NAME="E6:99"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <a href="newton.html#K13" title="NEWTON:func.13">SetPrimes</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T76" title="FINSEQ_1:th.76">FINSEQ_1:76</a></i>;<br/>
<a NAME="E7:99"/><b>then </b><i><font color="Green" title="E57">A3</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">SP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a></i>;<br/>
<a NAME="E8:99"/><i><font color="Green" title="E58">A4</font></i>: 
 <a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> <span class="p1">(<span class="default"><a href="newton.html#K15" title="NEWTON:func.15">primenumber</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">SP</font>
 
<b>by </b><i><a class="ref" href="newton.html#T82" target="_self" title="NEWTON:th.82">Th82</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">SPp</font> =  <a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> <span class="p1">(<span class="default"><a href="newton.html#K15" title="NEWTON:func.15">primenumber</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>;<br/>
<a NAME="E9:99"/><i><font color="Green" title="E59">A5</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> <span class="p2">(<span class="default"><a href="newton.html#K15" title="NEWTON:func.15">primenumber</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">SP</font>
 
<b>by </b><i><a class="txt" href="newton.html#E8:99"><i><font color="Green" title="E58">A4</font></i></a>, <a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a></i>;<br/>
<a NAME="E10:99"/>
<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="newton.html#K14" title="NEWTON:func.14">SetPrimenumber</a> <span class="p2">(<span class="default"><a href="newton.html#K15" title="NEWTON:func.15">primenumber</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="newton.html#D8" target="_self" title="NEWTON:def.8">Def8</a></i>;<br/>
<b>hence </b><a NAME="E11:99"/>
contradiction
 <b>by </b><i><a class="txt" href="newton.html#E7:99"><i><font color="Green" title="E57">A3</font></i></a>, <a class="txt" href="newton.html#E9:99"><i><font color="Green" title="E59">A5</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>


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