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<b>assume </b><a NAME="E1:106"/><i><font color="Green">E61</font></i>: 
 <a href="newton.html#K13" target="_self">SetPrimes</a>  is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>1</sub></font> =  <a href="newton.html#K13" target="_self">SetPrimes</a>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:106"/><i><font color="Green">E62</font></i>: 
 <a href="newton.html#K13" target="_self">SetPrimes</a> , <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R2">are_equipotent</a> 
 <b>by </b><i><a class="txt" href="newton.html#E1:106"><i><font color="Green">E61</font></i></a>, <a class="ref" href="finseq_1.html#T77">FINSEQ_1:77</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="newton.html#K15" target="_self">primenumber</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<a NAME="E5:106"/>
 <a href="card_1.html#K1">Card</a> <a href="newton.html#K13" target="_self">SetPrimes</a>  <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="newton.html#E4:106"><i><font color="Green">E62</font></i></a>, <a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/>
<a NAME="E6:106"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <a href="newton.html#K13" target="_self">SetPrimes</a>  <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T76">FINSEQ_1:76</a></i>;<br/>
<a NAME="E7:106"/><b>then </b><i><font color="Green">E63</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="card_1.html#D11">CARD_1:def 11</a></i>;<br/>
<a NAME="E8:106"/><i><font color="Green">E64</font></i>: 
 <a href="newton.html#K14" target="_self">SetPrimenumber</a> <span class="p1">(<span class="default"><a href="newton.html#K15" target="_self">primenumber</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="newton.html#T82" target="_self">Th82</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="newton.html#K14" target="_self">SetPrimenumber</a> <span class="p1">(<span class="default"><a href="newton.html#K15" target="_self">primenumber</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E10:106"/><i><font color="Green">E65</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="newton.html#E8:106"><i><font color="Green">E64</font></i></a>, <a class="ref" href="card_1.html#T80">CARD_1:80</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:106"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="newton.html#K15" target="_self">primenumber</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="int_2.html#V1">prime</a> ) </span>}</span>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="newton.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E13:106"/>
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="newton.html#E7:106"><i><font color="Green">E63</font></i></a>, <a class="txt" href="newton.html#E10:106"><i><font color="Green">E65</font></i></a>, <a class="txt" href="newton.html#E12:106"><i><font color="Green">E66</font></i></a>, <a class="ref" href="newton.html#T96" target="_self">Th96</a></i>;<br/>
<b>hence </b><a NAME="E14:106"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>


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