reserve Y for RealNormSpace;
reserve X,Y for RealBanachSpace;
reserve Z for open Subset of REAL;
reserve a,b,c,d,e,r,x0 for Real;
reserve y0 for VECTOR of X;
reserve G for Function of X,X;

theorem
a < b & Z = ].a,b.[ & G is_Lipschitzian_on the carrier of X implies
   ex y be continuous PartFunc of REAL,the carrier of X st
      dom y = ['a,b'] & y is_differentiable_on Z & y/.a = y0
    & for t be Real st t in Z holds diff(y,t) = G.(y/.t)
proof
   assume A1: a<b & Z = ].a,b.[
      & G is_Lipschitzian_on the carrier of X; then
   Fredholm(G,a,b,y0) is with_unique_fixpoint by Th57; then
   consider x be set such that
A2: x is_a_fixpoint_of Fredholm(G,a,b,y0)
  & for y being set st y is_a_fixpoint_of Fredholm(G,a,b,y0) holds x = y;
   x in dom Fredholm(G,a,b,y0) & x = (Fredholm(G,a,b,y0)).x by A2; then
   reconsider x as Element of the carrier of
     R_NormSpace_of_ContinuousFunctions(['a,b'],X);
   consider f be continuous PartFunc of REAL,the carrier of X such that
A4: x=f & dom f = ['a,b'] by ORDEQ_01:def 2;
   take f;
   thus dom f = ['a,b'] & f is_differentiable_on Z & f/.a = y0
      by Th58,A4,A1,A2;
A5:].a,b.[ c= [.a,b.] by XXREAL_1:25;
A6:['a,b'] = [.a,b.] by A1,INTEGRA5:def 3;
   let t be Real;
   assume A7:t in Z;
   dom G = the carrier of X by FUNCT_2:def 1; then
   rng f c= dom G; then
A8:dom (G*f) = ['a,b'] by A4,RELAT_1:27;
   thus diff(f,t) = (G*f)/.t by A7,Th58,A4,A1,A2
                 .= (G*f).t by A8,A5,A1,A7,A6,PARTFUN1:def 6
                 .= G.(f.t) by A8,A5,A1,A7,A6,FUNCT_1:12
                 .= G.(f/.t) by A5,A1,A7,A6,A4,PARTFUN1:def 6;
end;
