
theorem LS13:
  SetPrimenumber 13 = {2, 3, 5, 7, 11}
  proof
A1: {2, 3, 5, 7, 11} c= NAT
    proof
      let x be object;
      assume x in {2,3,5,7,11}; then
      x = 2 or x = 3 or x = 5 or x = 7 or x = 11 by ENUMSET1:def 3;
      hence thesis;
    end;
    for q being Nat holds q in {2,3,5,7,11} iff q < 13 & q is prime
    proof
      let q be Nat;
      hereby assume q in {2,3,5,7,11}; then
        q = 2 or q = 3 or q = 5 or q = 7 or q = 11 by ENUMSET1:def 3;
        hence q < 13 & q is prime by INT_2:28,PEPIN:41,59,NAT_4:26,NAT_4:27;
      end;
      assume
   Z: q < 13 & q is prime; then
      q < 12 + 1; then
      q <= 12 by NAT_1:13; then
      q = 0 or ... or q = 12;
      hence thesis
        by ENUMSET1:def 3,Z,INT_2:29,lem6,lem8,lem9,lem12,lem10;
    end;
    hence thesis by A1,NEWTON:def 7;
  end;
