
theorem
  for S, T being upper-bounded antisymmetric non empty RelStr, D being
Subset of [:S,T:] st [:S,T:] is complete or ex_inf_of D,[:S,T:] holds inf D = [
  inf proj1 D,inf proj2 D]
proof
  let S, T be upper-bounded antisymmetric non empty RelStr, D be Subset of [:S
  ,T:] such that
A1: [:S,T:] is complete or ex_inf_of D,[:S,T:];
  per cases;
  suppose
    D <> {};
    hence thesis by A1,YELLOW_3:47;
  end;
  suppose
A2: D = {};
    then
A3: inf proj2 D = Top T;
    inf D = Top [:S,T:] & inf proj1 D = Top S by A2;
    hence thesis by A3,Th3;
  end;
end;
