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

theorem Th12:
  for M be non empty MetrStruct for S be SetSequence of M for F be
  Subset-Family of TopSpaceMetr M st F = rng S holds ( S is open implies F is
  open ) & ( S is closed implies F is closed )
proof
  let M be non empty MetrStruct;
  let S be SetSequence of M;
  set T=TopSpaceMetr(M);
  let F be Subset-Family of T such that
A1: F = rng S;
  thus S is open implies F is open
  proof
    assume
A2: S is open;
    let P be Subset of T;
    assume P in F;
    then consider x being object such that
A3: x in dom S and
A4: S.x=P by A1,FUNCT_1:def 3;
    reconsider x as Nat by A3;
    S.x is open by A2;
    hence thesis by A4,Th6;
  end;
  assume
A5: S is closed;
  let P be Subset of T;
  assume P in F;
  then consider x being object such that
A6: x in dom S and
A7: S.x=P by A1,FUNCT_1:def 3;
  reconsider x as Nat by A6;
  S.x is closed by A5;
  hence thesis by A7,Th6;
end;
