reserve a,b,c,h for Integer;
reserve k,m,n for Nat;
reserve i,j,z for Integer;
reserve p for Prime;

theorem Th69:
  primeNumbers(0,10) = { 2,3,5,7 }
  proof
    thus primeNumbers(0,10) c= { 2,3,5,7 }
    proof
      let x be object;
      assume
A1:   x in primeNumbers(0,10);
      then x in seq(0,10);
      then consider k being Element of NAT such that
A2:   x = k and
A3:   1+0 <= k & k <= 0+10;
A4:   k is prime by A1,A2,NEWTON:def 6;
      k = 1 or ... or k = 10 by A3;
      hence thesis by A2,A4,ENUMSET1:def 2,XPRIMES0:4,6,8,9,10;
    end;
    let x be object;
    assume x in { 2,3,5,7 };
    then
A5: x = 2 or x = 3 or x = 5 or x = 7 by ENUMSET1:def 2;
    then
A6: x in SetPrimes by NEWTON:def 6,XPRIMES1:2,3,5,7;
    x in seq(0,10) by A5;
    hence thesis by A6,XBOOLE_0:def 4;
  end;
