
theorem
  for S, T being non empty reflexive antisymmetric RelStr st [:S,T:] is
  up-complete holds S is up-complete & T is up-complete
proof
  let S, T be non empty reflexive antisymmetric RelStr such that
A1: [:S,T:] is up-complete;
  thus S is up-complete
  proof
    set D = the non empty directed Subset of T;
    let X be non empty directed Subset of S;
    consider x being Element of [:S,T:] such that
A2: x is_>=_than [:X,D:] and
A3: for y being Element of [:S,T:] st y is_>=_than [:X,D:] holds x <=
    y by A1;
    take x`1;
    the carrier of [:S,T:] = [:the carrier of S, the carrier of T:] by
YELLOW_3:def 2;
    then
A4: x = [x`1,x`2] by MCART_1:21;
    hence x`1 is_>=_than X by A2,YELLOW_3:29;
    ex_sup_of [:X,D:],[:S,T:] by A1,WAYBEL_0:75;
    then ex_sup_of D,T by YELLOW_3:39;
    then
A5: sup D is_>=_than D by YELLOW_0:def 9;
    let y be Element of S;
    assume y is_>=_than X;
    then x <= [y,sup D] by A3,A5,YELLOW_3:30;
    hence thesis by A4,YELLOW_3:11;
  end;
  set D = the non empty directed Subset of S;
  let X be non empty directed Subset of T;
  consider x being Element of [:S,T:] such that
A6: x is_>=_than [:D,X:] and
A7: for y being Element of [:S,T:] st y is_>=_than [:D,X:] holds x <= y
  by A1;
  ex_sup_of [:D,X:],[:S,T:] by A1,WAYBEL_0:75;
  then ex_sup_of D,S by YELLOW_3:39;
  then
A8: sup D is_>=_than D by YELLOW_0:def 9;
  take x`2;
  the carrier of [:S,T:] = [:the carrier of S, the carrier of T:] by
YELLOW_3:def 2;
  then
A9: x = [x`1,x`2] by MCART_1:21;
  hence x`2 is_>=_than X by A6,YELLOW_3:29;
  let y be Element of T;
  assume y is_>=_than X;
  then x <= [sup D,y] by A7,A8,YELLOW_3:30;
  hence thesis by A9,YELLOW_3:11;
end;
