
theorem
  for S1, S2 being non empty reflexive RelStr for D1 being non empty
Subset of S1, D2 being non empty Subset of S2 st [:D1,D2:] is upper holds D1 is
  upper & D2 is upper
proof
  let S1, S2 be non empty reflexive RelStr, D1 be non empty Subset of S1, D2
  be non empty Subset of S2 such that
A1: [:D1,D2:] is upper;
  thus D1 is upper
  proof
    set q1 = the Element of D2;
    let x, y be Element of S1;
    assume that
A2: x in D1 and
A3: x <= y;
A4: [x,q1] in [:D1,D2:] by A2,ZFMISC_1:87;
    q1 <= q1;
    then [x,q1] <= [y,q1] by A3,Th11;
    then [y,q1] in [:D1,D2:] by A1,A4;
    hence thesis by ZFMISC_1:87;
  end;
  set q1 = the Element of D1;
  let x, y be Element of S2;
  assume that
A5: x in D2 and
A6: x <= y;
A7: [q1,x] in [:D1,D2:] by A5,ZFMISC_1:87;
  q1 <= q1;
  then [q1,x] <= [q1,y] by A6,Th11;
  then [q1,y] in [:D1,D2:] by A1,A7;
  hence thesis by ZFMISC_1:87;
end;
