reserve n for non zero Element of NAT;
reserve a,b,r,t for Real;
reserve
  Z for open Subset of REAL,
 y0 for VECTOR of REAL-NS n,
  G for Function of REAL-NS n,REAL-NS n;

theorem
  a<b & Z = ]. a,b .[ & G is_Lipschitzian_on the carrier of REAL-NS n implies
   ex y be continuous PartFunc of REAL,REAL-NS n 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 REAL-NS n; then
  Fredholm(G,a,b,y0) is with_unique_fixpoint by Th53; 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;
A3: x in dom (Fredholm(G,a,b,y0)) & x = (Fredholm(G,a,b,y0)).x
   by A2;
  reconsider x as Element of the carrier of
    R_NormSpace_of_ContinuousFunctions([' a,b '],REAL-NS n) by A3;
  consider f be continuous PartFunc of REAL,REAL-NS n such that
A4: x=f & dom f = [' a,b '] by Def2;
  take f;
  thus dom f = [' a,b '] & f is_differentiable_on Z & f/.a = y0
    by Th54,A4,A1,A3;
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 REAL-NS n 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,Th54,A4,A1,A3
                .= (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;
