reserve S, R for 1-sorted,
  X for Subset of R,
  T for TopStruct,
  x for set;
reserve H for non empty multMagma,
  P, Q, P1, Q1 for Subset of H,
  h for Element of H;
reserve G for Group,
  A, B for Subset of G,
  a for Element of G;

theorem Th27:
  for S, T being non empty TopSpace, f being Function of S, T, A
  being dense Subset of S st f is being_homeomorphism holds f.:A is dense
proof
  let S, T be non empty TopSpace, f be Function of S, T, A be dense Subset of
  S such that
A1: f is being_homeomorphism;
A2: rng f = [#]T by A1;
  Cl A = [#]S by TOPS_1:def 3;
  hence Cl (f.:A) = f.:the carrier of S by A1,TOPS_2:60
    .= [#]T by A2,RELSET_1:22;
end;
