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
  for E,F be non trivial RealBanachSpace,
        Z be Subset of E,
        f be PartFunc of E,F
  st Z is open & dom f = Z
   & f is_differentiable_on Z
   & f `| Z is_continuous_on Z
   & ( for x be Point of E st x in Z
       holds diff(f,x) is invertible )
  holds
    ( for x be Point of E,r1 be Real st x in Z & 0 < r1
      holds
        ex y be Point of F,r2 be Real
        st y = f.x & 0 < r2 & Ball(y,r2) c= f.: Ball(x,r1) )
    & f.:Z is open
  proof
    let E,F be non trivial RealBanachSpace,
          Z be Subset of E,
          f be PartFunc of E,F;
    assume
    A1: Z is open & dom f = Z
      & f is_differentiable_on Z
      & f `| Z is_continuous_on Z;

    assume
    A2: for x be Point of E st x in Z
        holds diff(f,x) is invertible;

    thus
    A3: for x be Point of E,r1 be Real st x in Z & 0 < r1
        holds
          ex y be Point of F,r2 be Real
          st y = f.x & 0 < r2 & Ball(y,r2) c= f.: Ball(x,r1)
    proof
      let x be Point of E,r1 be Real;
      assume
      A4: x in Z & 0 < r1;
      f.x in rng f by A1,A4,FUNCT_1:3; then
      reconsider y = f.x as Point of F;
      diff(f,x) is invertible by A2,A4; then
      consider r2 be Real such that
      A5: 0 < r2 & Ball(y,r2) c= f.: Ball(x,r1) by A1,A4,Th18;

      take y,r2;
      thus thesis by A5;
    end;

    for y be Point of F st y in f.:Z
    ex r be Real st 0 < r & Ball(y,r) c= f.:Z
    proof
      let y be Point of F;
      assume y in f.:Z; then
      consider x be object such that
      A6: x in dom f & x in Z & y = f.x by FUNCT_1:def 6;
      reconsider x as Point of E by A6;

      consider r1 be Real such that
      A7: 0 < r1 & Ball(x,r1) c= Z by A1,A6,NDIFF_8:20;

      consider y1 be Point of F,r2 be Real such that
      A8: y1 = f.x & 0 < r2 & Ball(y1,r2) c= f.: Ball(x,r1) by A3,A6,A7;

      take r2;
      thus 0 < r2 by A8;

      f.: Ball(x,r1) c= f.:Z by A7,RELAT_1:123;
      hence Ball(y,r2) c= f.:Z by A6,A8,XBOOLE_1:1;
    end;

    hence f.:Z is open by NDIFF_8:20;
  end;
