reserve E, x, y, X for set;
reserve A, B, C for Subset of E^omega;
reserve a, b for Element of E^omega;
reserve i, k, l, kl, m, n, mn for Nat;

theorem
  m <= n & k <= l & A c= C |^ (m, n) & B c= C |^ (k, l) implies A ^^ B
  c= C |^ (m + k, n + l)
proof
  assume that
A1: m <= n & k <= l and
A2: A c= C |^ (m, n) & B c= C |^ (k, l);
  thus thesis
  proof
    let x be object;
    assume x in A ^^ B;
    then consider a, b such that
A3: a in A & b in B and
A4: x = a ^ b by FLANG_1:def 1;
    a ^ b in C |^ (m, n) ^^ C |^ (k, l) by A2,A3,FLANG_1:def 1;
    hence thesis by A1,A4,Th37;
  end;
end;
