reserve E, x, y, X for set;
reserve A, B, C, D for Subset of E^omega;
reserve a, a1, a2, b, c, c1, c2, d, ab, bc for Element of E^omega;
reserve e for Element of E;
reserve i, j, k, l, n, n1, n2, m for Nat;

theorem
  (A ^^ B) \+\ (A ^^ C) c= A ^^ (B \+\ C) & (B ^^ A) \+\ (C ^^ A) c= (B
  \+\ C) ^^ A
proof
  (A ^^ B) \+\ (A ^^ C) = ((A ^^ B) \/ (A ^^ C)) \ ((A ^^ B) /\ (A ^^ C))
  by XBOOLE_1:101
    .= (A ^^ (B \/ C)) \ ((A ^^ B) /\ (A ^^ C)) by Th20;
  then
A1: (A ^^ B) \+\ (A ^^ C) c= (A ^^ (B \/ C)) \ (A ^^ (B /\ C)) by Th19,
XBOOLE_1:34;
  (B ^^ A) \+\ (C ^^ A) = ((B ^^ A) \/ (C ^^ A)) \ ((B ^^ A) /\ (C ^^ A))
  by XBOOLE_1:101
    .= ((B \/ C) ^^ A) \ ((B ^^ A) /\ (C ^^ A)) by Th20;
  then
A2: (B ^^ A) \+\ (C ^^ A) c= ((B \/ C) ^^ A) \ ((B /\ C) ^^ A) by Th19,
XBOOLE_1:34;
  ((B \/ C) ^^ A) \ ((B /\ C) ^^ A) c= ((B \/ C) \ (B /\ C)) ^^ A by Th21;
  then
A3: (B ^^ A) \+\ (C ^^ A) c= ((B \/ C) \ (B /\ C)) ^^ A by A2;
  (A ^^ (B \/ C)) \ (A ^^ (B /\ C)) c= A ^^ ((B \/ C) \ (B /\ C)) by Th21;
  then (A ^^ B) \+\ (A ^^ C) c= A ^^ ((B \/ C) \ (B /\ C)) by A1;
  hence thesis by A3,XBOOLE_1:101;
end;
