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 Th58:
for f,g be continuous PartFunc of REAL,the carrier of X
  st dom f =['a,b'] & dom g =['a,b'] & Z = ].a,b.[
   & a<b & G is_Lipschitzian_on the carrier of X
   & 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,the carrier of X;
   assume A1: dom f =['a,b'] & dom g =['a,b'] & Z = ].a,b.[
           & a<b & G is_Lipschitzian_on the carrier of X
           & g=Fredholm(G,a,b,y0).f;
X1:['a,b'] = [.a,b.] by A1,INTEGRA5:def 3;
   set D = R_NormSpace_of_ContinuousFunctions(['a,b'],X);
X2:dom G = the carrier of X by FUNCT_2:def 1; then
A2:G is_continuous_on dom G by A1,NFCONT_1:45;
   f is Element of D by ORDEQ_01:def 2,A1; then
   consider f0,g0,Gf0 be continuous PartFunc of REAL,the carrier of X
     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,Def8;
   reconsider Gf=G*f as continuous PartFunc of REAL,the carrier of X by A3;
   rng f c= dom G by X2; then
   dom Gf =['a,b'] by A1,RELAT_1:27;
   hence thesis by A1,X1,Th40,Th40a,A3;
end;
