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

theorem Th7:
  for M be MetrStruct for S be SetSequence of M, S9 be SetSequence
of TopSpaceMetr M st S9 = S holds ( S is open iff S9 is open ) & ( S is closed
  iff S9 is closed )
proof
  let M be MetrStruct;
  let S be SetSequence of M, S9 be SetSequence of TopSpaceMetr M such that
A1: S9 = S;
  thus S is open implies S9 is open
  proof
    assume
A2: S is open;
    let i;
    S.i is open by A2;
    hence thesis by A1,Th6;
  end;
  thus S9 is open implies S is open
  proof
    assume
A3: S9 is open;
    let i;
    S9.i is open by A3;
    hence thesis by A1,Th6;
  end;
  thus S is closed implies S9 is closed
  proof
    assume
A4: S is closed;
    let i;
    S.i is closed by A4;
    hence thesis by A1,Th6;
  end;
  assume
A5: S9 is closed;
  let i;
  S9.i is closed by A5;
  hence thesis by A1,Th6;
end;
