
theorem
  for A, B, C, D being set st A meets B & C meets D
   holds [: A, C :] meets [: B, D :]
proof
  let A, B, C, D be set;
  assume that
A1: A meets B and
A2: C meets D;
  consider x being object such that
A3: x in A & x in B by A1,XBOOLE_0:3;
  consider y being object such that
A4: y in C & y in D by A2,XBOOLE_0:3;
  [x,y] in [: A, C :] & [x,y] in [: B, D :] by A3,A4,ZFMISC_1:87;
  hence thesis by XBOOLE_0:3;
end;
