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 Th18:
  for E,F be non trivial RealBanachSpace,
      Z be Subset of E,
      f be PartFunc of E,F,
      a be Point of E,
      b be Point of F
  st Z is open & dom f = Z
   & f is_differentiable_on Z
   & f `| Z is_continuous_on Z
   & a in Z & f.a = b
   & diff(f,a) is invertible
  holds
   ( for r1 be Real st 0 < r1
     holds
       ex r2 be Real
       st 0 < r2 & Ball(b,r2) c= f.: Ball(a,r1) )
  proof
    let E,F be non trivial RealBanachSpace,
          Z be Subset of E,
          f be PartFunc of E,F,
          a be Point of E,
          b be Point of F;

    assume
    A1: Z is open & dom f = Z
       & f is_differentiable_on Z
       & f `| Z is_continuous_on Z
       & a in Z & f.a = b
       & diff(f,a) is invertible;

    then
    consider A be Subset of E,B be Subset of F,
             g be PartFunc of F,E such that
    A2: A is open & B is open
      & A c= dom f
      & a in A & b in B
      & f.:A = B
      & dom g = B & rng g = A
      & dom(f|A) = A & rng(f|A) = B
      & (f|A) is one-to-one
      & g is one-to-one
      & g = (f|A)"
      & (f|A) = g"
      & g.b = a
      & g is_continuous_on B
      & g is_differentiable_on B
      & g `| B is_continuous_on B
      & ( for y be Point of F st y in B
          holds diff(f,g/.y) is invertible )
      & ( for y be Point of F st y in B
          holds diff(g,y) = Inv diff(f,g/.y) ) by Th17;

    let r1 be Real;
    assume
    A3: 0 < r1;

    A5: g"Ball(a,r1) = (f|A).:(Ball(a,r1)) by A2,FUNCT_1:85;
    A6: (f|A).:(Ball(a,r1)) c= f.:(Ball(a,r1)) by RELAT_1:128;

    a in dom (f|A) & a in Ball(a,r1) & f.a = (f|A).a
      by A2,A3,FUNCT_1:49,NDIFF_8:13; then
    f.a in (f|A).:(Ball(a,r1)) by FUNCT_1:def 6;

    then
    consider r2 be Real such that
    A7: 0 < r2 & Ball(b,r2) c= (f|A).:(Ball(a,r1))
      by A1,A2,A5,Th1,NDIFF_8:20;

    thus thesis by A6,A7,XBOOLE_1:1;
  end;
