
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 sups-inheriting implies S2 is 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;
  assume
A2: the carrier of S1 = the carrier of S2;
  assume
A3: for X being Subset of S1 st ex_sup_of X,T1
  holds "\/"(X,T1) in the carrier of S1;
  let X be Subset of S2;
  reconsider Y = X as Subset of S1 by A2;
  assume
A4: ex_sup_of X,T2;
  then "\/"(Y,T1) in the carrier of S1 by A1,A3,YELLOW_0:14;
  hence thesis by A1,A2,A4,YELLOW_0:26;
end;
