
theorem Th27:
  for S, T being non empty TopSpace st the TopStruct of S = the
  TopStruct of T & S is monotone-convergence holds T is monotone-convergence
proof
  let S, T be non empty TopSpace such that
A1: the TopStruct of S = the TopStruct of T and
A2: for D being non empty directed Subset of Omega S holds ex_sup_of D,
  Omega S & for V being open Subset of S st sup D in V holds D meets V;
  let E be non empty directed Subset of Omega T;
A3: Omega S = Omega T by A1,Th13;
  hence ex_sup_of E,Omega T by A2;
  let V be open Subset of T such that
A4: sup E in V;
  reconsider W = V as Subset of S by A1;
  W is open by A1,TOPS_3:76;
  hence thesis by A2,A3,A4;
end;
