reserve r, s, t, g for Real,

          r3, r1, r2, q3, p3 for Real;
reserve T for TopStruct,
  f for RealMap of T;

theorem
  for R being Subset-Family of REAL st f is continuous & R is open holds
  ("f).:R is open
proof
  let R be Subset-Family of REAL;
  assume
A1: f is continuous;
  assume
A2: R is open;
  let P be Subset of T;
  assume P in ("f).:R;
  then consider eR being object such that
A3: eR in bool REAL and
A4: eR in R and
A5: P = ("f).eR by FUNCT_2:64;
   reconsider eR as set by TARSKI:1;
A6: P = f"eR by A3,A5,MEASURE6:def 3;
  reconsider eR as Subset of REAL by A3;
  eR is open by A2,A4;
  hence thesis by A1,A6,Th8;
end;
