reserve i,n,m for Nat,
  x,y,X,Y for set,
  r,s for Real;

theorem Th6:
  for M be MetrStruct for A be Subset of M, A9 be Subset of
TopSpaceMetr M st A9 = A holds ( A is open iff A9 is open ) & ( A is closed iff
  A9 is closed )
proof
  let M be MetrStruct;
  let A be Subset of M, A9 be Subset of TopSpaceMetr M such that
A1: A9 = A;
  thus A is open implies A9 is open
  by A1,PRE_TOPC:def 2;
  thus A9 is open implies A is open
  by PRE_TOPC:def 2,A1;
  thus A is closed implies A9 is closed
  proof
    assume A is closed;
    then A` is open;
    then A` in Family_open_set M;
    then A9` is open by A1,PRE_TOPC:def 2;
    hence thesis by TOPS_1:3;
  end;
  assume A9 is closed;
  then A` in Family_open_set M by A1,PRE_TOPC:def 2;
  then A` is open;
  hence thesis;
end;
