
theorem Th47:
  for S being up-complete antisymmetric non empty reflexive RelStr,
  T being non empty reflexive RelStr st the RelStr of S = the RelStr of T
  for A being Subset of S, C being Subset of T st A = C & A is inaccessible
  holds C is inaccessible
proof
  let S be up-complete antisymmetric non empty reflexive RelStr,
  T be non empty reflexive RelStr such that
A1: the RelStr of S = the RelStr of T;
  let A be Subset of S, C be Subset of T such that
A2: A = C and
A3: for D being non empty directed Subset of S st sup D in A holds D meets A;
  let D be non empty directed Subset of T such that
A4: sup D in C;
  reconsider E = D as non empty directed Subset of S by A1,WAYBEL_0:3;
  ex_sup_of E,S by WAYBEL_0:75;
  then sup D = sup E by A1,YELLOW_0:26;
  hence thesis by A2,A3,A4;
end;
