reserve i,j,k,n,m,l,s,t for Nat;
reserve a,b for Real;
reserve F for real-valued FinSequence;
reserve z for Complex;
reserve x,y for Complex;
reserve r,s,t for natural Number;
reserve p,q for natural Number;
reserve i0,i,i1,i2,i4 for Integer;
reserve x for set;
reserve p for Prime;

theorem Th79:
  SetPrimes is infinite
proof
  assume
A1: SetPrimes is finite;
  then reconsider SP = SetPrimes as finite set;
  consider n be Nat such that
A2: SetPrimes,Seg n are_equipotent by A1,FINSEQ_1:56;
  card SetPrimes = card Seg n by A2,CARD_1:5;
  then card SetPrimes = card n by FINSEQ_1:55;
  then
A3: card SP = n;
  set p=primenumber (n+1);
  set SPp = SetPrimenumber p;
A4: n+1 = card SetPrimenumber p by Def8;
  card SPp <= card SP by Th68,NAT_1:43;
  hence contradiction by A3,A4,NAT_1:13;
end;
