
theorem
  for S, T being up-complete non empty Poset, X being Subset of S, Y
  being Subset of T st X is Open & Y is Open holds [:X,Y:] is Open
proof
  let S, T be up-complete non empty Poset, X be Subset of S, Y be Subset of
  T such that
A1: for x being Element of S st x in X ex y being Element of S st y in X
  & y << x and
A2: for x being Element of T st x in Y ex y being Element of T st y in Y
  & y << x;
  let x be Element of [:S,T:];
  assume
A3: x in [:X,Y:];
  then
A4: x = [x`1,x`2] by MCART_1:21;
  then x`1 in X by A3,ZFMISC_1:87;
  then consider s being Element of S such that
A5: s in X and
A6: s << x`1 by A1;
  x`2 in Y by A3,A4,ZFMISC_1:87;
  then consider t being Element of T such that
A7: t in Y and
A8: t << x`2 by A2;
  reconsider t as Element of T;
  take [s,t];
  thus [s,t] in [:X,Y:] by A5,A7,ZFMISC_1:87;
  thus thesis by A4,A6,A8,Th19;
end;
