
theorem
  for T1,T2 being non empty RelStr for S1 being non empty full SubRelStr of T1
  for S2 being non empty full SubRelStr of T2
  st the RelStr of T1 = the RelStr of T2 &
  the carrier of S1 = the carrier of S2
  holds S1 is directed-sups-inheriting implies S2 is directed-sups-inheriting
proof
  let T1,T2 be non empty RelStr;
  let S1 be non empty full SubRelStr of T1;
  let S2 be non empty full SubRelStr of T2;
  assume
A1: the RelStr of T1 = the RelStr of T2;
  the RelStr of S2 = the RelStr of S2;
  then reconsider R = S2 as full SubRelStr of T1 by A1,Th12;
  assume
A2: the carrier of S1 = the carrier of S2;
  then
A3: the RelStr of S1 = the RelStr of R by YELLOW_0:57;
  assume
A4: for X being directed Subset of S1 st X <> {} & ex_sup_of X,T1
  holds "\/"(X,T1) in the carrier of S1;
  let X be directed Subset of S2 such that
A5: X <> {};
  reconsider Y = X as directed Subset of S1 by A3,WAYBEL_0:3;
  assume
A6: ex_sup_of X,T2;
  then "\/"(Y,T1) in the carrier of S1 by A1,A4,A5,YELLOW_0:14;
  hence thesis by A1,A2,A6,YELLOW_0:26;
end;
