
theorem
  for X, Y being non empty lower-bounded antisymmetric RelStr st [:X,Y:]
  is complete holds X is complete & Y is complete
proof
  let X, Y be non empty lower-bounded antisymmetric RelStr such that
A1: [:X,Y:] is complete;
  for A being Subset of X holds ex_sup_of A,X
  proof
    let A be Subset of X;
    per cases;
    suppose
A2:   A is non empty;
      set B = the non empty Subset of Y;
      ex_sup_of [:A,B:],[:X,Y:] by A1,YELLOW_0:17;
      hence thesis by A2,Th39;
    end;
    suppose
      A is empty;
      hence thesis by YELLOW_0:42;
    end;
  end;
  hence X is complete by YELLOW_0:53;
  for B being Subset of Y holds ex_sup_of B,Y
  proof
    let B be Subset of Y;
    per cases;
    suppose
A3:   B is non empty;
      set A = the non empty Subset of X;
      ex_sup_of [:A,B:],[:X,Y:] by A1,YELLOW_0:17;
      hence thesis by A3,Th39;
    end;
    suppose
      B is empty;
      hence thesis by YELLOW_0:42;
    end;
  end;
  hence thesis by YELLOW_0:53;
end;
