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 Th59:
for y be continuous PartFunc of REAL,the carrier of X
  st a < b & Z = ].a,b.[ & G is_Lipschitzian_on the carrier of X
   & 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,the carrier of X;
   assume A1: a<b & Z = ].a,b.[ & G is_Lipschitzian_on the carrier of X
        & 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));
X1:['a,b'] = [.a,b.] by A1,INTEGRA5:def 3;
A2:dom (Fredholm(G,a,b,y0))
     = the carrier of R_NormSpace_of_ContinuousFunctions(['a,b'],X)
         by FUNCT_2:def 1;
A3:y is Element of the carrier of
    R_NormSpace_of_ContinuousFunctions(['a,b'],X) by ORDEQ_01:def 2,A1;
X2:dom G = the carrier of X 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,the carrier of X
      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 Def8,A1,A3;
   rng f c= dom G by X2; 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;
    hence diff(y,t) = G.(y/.t) by A1
                  .= G.(y.t) by A1,A7,PARTFUN1:def 6
                  .= (Gf).t by A5,A1,A7,FUNCT_1:13
                  .= (Gf)/.t by A5,A1,A7,A6,PARTFUN1:def 6;
   end;
   hence thesis by A1,X1,A2,A5,Th47,A6,ORDEQ_01:def 2;
end;
