reserve S,T,W,Y for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem Th1:
  for X,Y be RealNormSpace,
      f be PartFunc of X,Y,
      A be Subset of X,
      B be Subset of Y
  st dom f = A
   & f is_continuous_on A
   & A is open
   & B is open
  holds f"B is open
  proof
    let X,Y be RealNormSpace,
          f be PartFunc of X,Y,
          A be Subset of X,
          B be Subset of Y;
    assume that
    A1: dom f = A and
    A2: f is_continuous_on A and
    A3: A is open and
    A4: B is open;

    for a being Point of X st a in f"B holds
    ex s being Real st
    s > 0 & Ball(a,s) c= f"B
    proof
      let a be Point of X;
      assume a in f"B; then
      A6: a in dom f & f.a in B by FUNCT_1:def 7; then
      f/.a in B by PARTFUN1:def 6; then
      consider t be Real such that
      A7: t > 0 & Ball(f/.a, t) c= B by A4,NDIFF_8:20;
      consider s0 being Real such that
      A8: 0 < s0
        & for a1 being Point of X st a1 in A & ||.(a1 - a).|| < s0 holds
        ||.(f /. a1) - (f /. a).|| < t by A1,A2,A6,A7,NFCONT_1:19;

      consider s1 be Real such that
      A9: 0 < s1 & Ball(a,s1) c= A by A1,A3,A6,NDIFF_8:20;
      set s = min(s0,s1);

      A10: s <= s0 by XXREAL_0:17;
      A11: s <= s1 by XXREAL_0:17;
      take s;
      thus 0 < s by A8,A9,XXREAL_0:15;
      for a1 be object st a1 in Ball(a,s) holds a1 in f"B
      proof
        let a1 be object;
        assume
        A13: a1 in Ball(a,s); then
        reconsider a1 as Point of X;
        A14: a1 in Ball(a,s1) by A11,A13,NDIFF_8:15,TARSKI:def 3;
        A15: a1 in Ball(a,s0) by A10,A13,NDIFF_8:15,TARSKI:def 3;
        a1 in { x where x is Point of X : ||.(x - a).|| < s0 }
          by A15,NDIFF_8:17; then
        ex x be Point of X st a1 = x & ||.(x - a).|| < s0; then
        ||.(f /. a1) - (f /. a).|| < t by A8,A9,A14; then
        f /. a1 in { y where y is Point of Y : ||.(y - f/.a).|| < t }; then
        f /. a1 in Ball(f/.a, t) by NDIFF_8:17; then
        f . a1 in Ball(f/.a, t) by A1,A9,A14,PARTFUN1:def 6;
        hence thesis by A1,A7,A9,A14,FUNCT_1:def 7;
      end;
      hence Ball(a,s) c= f"B by TARSKI:def 3;
    end;
    hence f"B is open by NDIFF_8:20;
  end;
