 reserve U for set,
         X, Y for Subset of U;

theorem Th6:
  for A, B, C, D being Subset of U st
    Inter (A, B) <> {} & Inter (A, B) = Inter (C, D) holds
      A = C & B = D
  proof
    let A,B,C,D be Subset of U;
    assume A1: Inter (A,B) <> {} & Inter (A,B) = Inter (C,D);
    then A in Inter (A,B) by Th2; then
A2: C c= A & A c= D by Th1,A1;
    C in Inter (C,D) by A1,Th2; then
A3: A c= C & C c= B by A1,Th1;
    B in Inter (A,B) by A1,Th2; then
A4: B c= D by Th1,A1;
    D in Inter (C,D) by A1,Th2; then
    D c= B by A1,Th1;
    hence thesis by A3,A4,A2;
  end;
