reserve T for non empty TopSpace,
  A, B for Subset of T,
  F, G for Subset-Family of T;
reserve x for Point of T;

theorem Th52:
  A c= B implies A^0 c= B^0
proof
  assume
A1: A c= B;
  let x be object;
  assume
A2: x in A^0;
  then reconsider x9 = x as Point of T;
  x9 is_a_condensation_point_of A by A2,Def10;
  then x9 is_a_condensation_point_of B by A1,Th48;
  hence thesis by Def10;
end;
