
theorem LS4:
  SetPrimenumber 7 = {2, 3, 5}
  proof
A1: {2, 3, 5} c= NAT
    proof
      let x be object;
      assume x in {2,3,5}; then
      x = 2 or x = 3 or x = 5 by ENUMSET1:def 1;
      hence thesis;
    end;
    for q being Nat holds q in {2,3,5} iff q < 7 & q is prime
    proof
      let q be Nat;
      hereby assume q in {2,3,5}; then
        q = 2 or q = 3 or q = 5 by ENUMSET1:def 1;
        hence q < 7 & q is prime by INT_2:28,PEPIN:41,59;
      end;
      assume
   Z: q < 7 & q is prime; then
      q < 6 + 1; then
      q <= 6 by NAT_1:13; then
      q = 0 or ... or q = 6;
      hence thesis by ENUMSET1:def 1,Z,INT_2:29,lem6;
    end;
    hence thesis by A1,NEWTON:def 7;
  end;
