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 Th54:
  for f,g be continuous PartFunc of REAL,REAL-NS n
    st dom f =[' a,b '] & dom g =[' a,b '] & Z = ]. a,b .[
     & a<b
     & G is_Lipschitzian_on the carrier of REAL-NS n
     & g=Fredholm(G,a,b,y0).f holds g/.a=y0
      & g is_differentiable_on Z
      & for t be Real st t in Z holds diff(g,t) = (G*f)/.t
proof
  let f,g be continuous PartFunc of REAL,REAL-NS n;
  assume A1: dom f =[' a,b '] & dom g =[' a,b '] & Z = ]. a,b .[
           & a<b
           & G is_Lipschitzian_on the carrier of REAL-NS n
           & g=Fredholm(G,a,b,y0).f;
  set D = R_NormSpace_of_ContinuousFunctions([' a,b '],REAL-NS n);
  dom G = the carrier of REAL-NS n by FUNCT_2:def 1; then
A2: G is_continuous_on dom G by A1,NFCONT_1:45;
  f is Element of D by Def2,A1; then
  consider f0,g0,Gf0 be continuous PartFunc of REAL,REAL-NS n such that
A3: f=f0 & Fredholm(G,a,b,y0).f = g0
  & dom f0 = [' a,b '] & dom g0 =[' a,b '] & Gf0 = G*f0
  & for t be Real st t in [' a,b ']
      holds g0.t = y0+ integral(Gf0,a,t) by A1,A2,Def7;
  reconsider Gf=G*f as continuous PartFunc of REAL,REAL-NS n by A3;
  dom G = the carrier of REAL-NS n by FUNCT_2:def 1; then
  rng f c= dom G; then
  dom Gf =[' a,b ']  by A1,RELAT_1:27;
  hence thesis by A1,Th36,A3;
end;
