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

theorem
  primeNumbers(3,10) = { 5,7,11,13 }
  proof
    thus primeNumbers(3,10) c= { 5,7,11,13 }
    proof
      let x be object;
      assume
A1:   x in primeNumbers(3,10);
      then x in seq(3,10);
      then consider k being Element of NAT such that
A2:   x = k and
A3:   1+3 <= k & k <= 3+10;
      k = 4 or ... or k = 13 by A3;
      hence thesis by A1,A2,ENUMSET1:def 2,XPRIMES0:4,6,8,9,10,12,NEWTON:def 6;
    end;
    let x be object;
    assume x in { 5,7,11,13 };
    then
A4: x = 5 or x = 7 or x = 11 or x = 13 by ENUMSET1:def 2;
    then
A5: x in SetPrimes by NEWTON:def 6,XPRIMES1:5,7,11,13;
    x in seq(3,10) by A4;
    hence thesis by A5,XBOOLE_0:def 4;
  end;
