
theorem LS11:
  SetPrimenumber 11 = {2, 3, 5, 7}
  proof
A1: {2, 3, 5, 7} c= NAT
    proof
      let x be object;
      assume x in {2,3,5,7}; then
      x = 2 or x = 3 or x = 5 or x = 7 by ENUMSET1:def 2;
      hence thesis;
    end;
    for q being Nat holds q in {2,3,5,7} iff q < 11 & q is prime
    proof
      let q be Nat;
      hereby assume q in {2,3,5,7}; then
        q = 2 or q = 3 or q = 5 or q = 7 by ENUMSET1:def 2;
        hence q < 11 & q is prime by INT_2:28,PEPIN:41,59,NAT_4:26;
      end;
      assume
   Z: q < 11 & q is prime; then
      q < 10 + 1; then
      q <= 10 by NAT_1:13; then
      q = 0 or ... or q = 10;
      hence thesis by ENUMSET1:def 2,Z,INT_2:29,lem6,lem8,lem9,lem10;
    end;
    hence thesis by A1,NEWTON:def 7;
  end;
