
theorem Th29:
  for S1, S2 being non empty RelStr for D1 being non empty Subset
  of S1, D2 being non empty Subset of S2 for x being Element of S1, y being
  Element of S2 st [x,y] is_>=_than [:D1,D2:] holds x is_>=_than D1 & y
  is_>=_than D2
proof
  let S1, S2 be non empty RelStr, D1 be non empty Subset of S1, D2 be non
  empty Subset of S2, x be Element of S1, y be Element of S2 such that
A1: [x,y] is_>=_than [:D1,D2:];
  thus x is_>=_than D1
  proof
    set a = the Element of D2;
    let b be Element of S1;
    assume b in D1;
    then [b,a] in [:D1,D2:] by ZFMISC_1:87;
    then [b,a] <= [x,y] by A1;
    hence thesis by Th11;
  end;
  set b = the Element of D1;
  let a be Element of S2;
  assume a in D2;
  then [b,a] in [:D1,D2:] by ZFMISC_1:87;
  then [b,a] <= [x,y] by A1;
  hence thesis by Th11;
end;
