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<b>assume </b><a NAME="E1:102"/><i><font color="Green">E41</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">SP</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">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:102"/><i><font color="Green">E42</font></i>: 
 <a href="newton.html#K13" target="_self">SetPrimes</a> , <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> <a href="tarski.html#R2">are_equipotent</a> 
 <b>by </b><i><a class="ref" href="newton.html#T9" target="_self">Th9</a>, <a class="ref" href="finseq_1.html#T77">FINSEQ_1:77</a></i>;<br/>
<b>set </b><font color="Maroon">p</font> =  <a href="newton.html#K15" target="_self">primenumber</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<a NAME="E5:102"/>
 <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">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="newton.html#T10" target="_self">Th10</a>, <a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/>
<a NAME="E6:102"/><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">n</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T76">FINSEQ_1:76</a></i>;<br/>
<a NAME="E7:102"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">SP</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="card_1.html#D11">CARD_1:def 11</a></i>;<br/>
<a NAME="E8:102"/><i><font color="Green">E46</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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">SP</font>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">SPp</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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>;<br/>
<a NAME="E9:102"/><i><font color="Green">E47</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="newton.html#K14" target="_self">SetPrimenumber</a> <span class="p2">(<span class="default"><a href="newton.html#K15" target="_self">primenumber</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">SP</font>
 
<b>by </b><i>, <a class="ref" href="card_1.html#T80">CARD_1:80</a></i>;<br/>
<a NAME="E10:102"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="newton.html#K14" target="_self">SetPrimenumber</a> <span class="p2">(<span class="default"><a href="newton.html#K15" target="_self">primenumber</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E11:102"/>
contradiction
 <b>by </b><i><a class="ref" href="newton.html#T11" target="_self">Th11</a>, <a class="ref" href="newton.html#T15" target="_self">Th15</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>


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