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 Th55:
  for y be continuous PartFunc of REAL,REAL-NS n st a<b
     & Z = ]. a,b .[
     & G is_Lipschitzian_on the carrier of REAL-NS n
     & 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))
   holds y is_a_fixpoint_of (Fredholm(G,a,b,y0))
proof
  let y be continuous PartFunc of REAL,REAL-NS n;
  assume A1: a<b & Z = ]. a,b .[
        & G is_Lipschitzian_on the carrier of REAL-NS n
        & 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));
A2: dom (Fredholm(G,a,b,y0))
  = the carrier of R_NormSpace_of_ContinuousFunctions([' a,b '],REAL-NS n)
  by FUNCT_2:def 1;
A3: y is Element of the carrier of
  R_NormSpace_of_ContinuousFunctions([' a,b '],REAL-NS n) by Def2,A1;
A4: y in dom (Fredholm(G,a,b,y0)) by A2,Def2,A1;
  dom G = the carrier of REAL-NS n by FUNCT_2:def 1; then
  G is_continuous_on dom G by A1,NFCONT_1:45; then
  consider f,g,Gf be continuous PartFunc of REAL,REAL-NS n such that
A5: y=f & (Fredholm(G,a,b,y0)).y = g
  & dom f =[' a,b '] & dom g =[' a,b '] & Gf = G*f
  & for t be Real st t in [' a,b ']
      holds g.t = y0+ integral(Gf,a,t) by Def7,A1,A3;
  dom G = the carrier of REAL-NS n  by FUNCT_2:def 1; then
  rng f c= dom G; then
  A6: dom (G*f) = [' a,b '] by A5,RELAT_1:27;
  for t be Real st t in Z holds diff(y,t) = (Gf)/.t
  proof
    let t be Real;
    assume A7:t in Z;
  A8: ].a,b.[ c= [.a,b.] by XXREAL_1:25;
  A9: ['a,b'] = [.a,b.] by A1,INTEGRA5:def 3;
    thus diff(y,t) = G.(y/.t) by A1,A7
                  .= G.(y.t) by A8,A1,A7,A9,PARTFUN1:def 6
                  .= (Gf).t by A5,A8,A1,A7,A9,FUNCT_1:13
                  .= (Gf)/.t by A5,A8,A1,A7,A9,A6,PARTFUN1:def 6;
  end;
  hence thesis by A4,A5,Th43,A1,A6;
end;
