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 Th55:
for m be Nat st a <= b & 0 < r
 & (for y1,y2 be VECTOR of X holds ||.G/.y1-G/.y2.|| <= r*||.y1-y2.||) holds
  for u,v be VECTOR of R_NormSpace_of_ContinuousFunctions(['a,b'],X)
    holds ||. iter(Fredholm(G,a,b,y0),(m+1)).u
              - iter(Fredholm(G,a,b,y0),(m+1)).v .||
                     <= ((r*(b-a))|^(m+1))/((m+1)!) * ||.u-v.||
proof
   let m be Nat;
   assume A1: a<=b & 0 < r
     & for y1,y2 be VECTOR of X holds ||.G/.y1-G/.y2.||<=r*||.y1-y2.||;
   let u,v be VECTOR of
      R_NormSpace_of_ContinuousFunctions(['a,b'],X);
   reconsider u1=iter(Fredholm(G,a,b,y0),(m+1)).u as VECTOR of
      R_NormSpace_of_ContinuousFunctions(['a,b'],X);
   reconsider v1=iter(Fredholm(G,a,b,y0),(m+1)).v as VECTOR of
      R_NormSpace_of_ContinuousFunctions(['a,b'],X);
   consider g be continuous PartFunc of REAL,the carrier of X such that
A2: u1=g & dom g = ['a,b'] by ORDEQ_01:def 2;
   consider h be continuous PartFunc of REAL,the carrier of X such that
A3: v1=h & dom h = ['a,b'] by ORDEQ_01:def 2;
   now let t be Real;
A4: ['a,b'] = [.a,b.] by A1,INTEGRA5:def 3;
    assume A5: t in ['a,b']; then
A6: ex g be Real st t=g & a<=g & g <= b by A4;
    m in NAT by ORDINAL1:def 12; then
A7: ||. g/.t - h/.t .||
        <= ((r*(t-a))|^(m+1) )/((m+1)!) * ||.u-v.|| by A5,Th54,A1,A2,A3;
    ((r*(t-a))|^(m+1) )/((m+1)!) * ||.u-v.||
    <= ((r*(b-a))|^(m+1) )/((m+1)!) * ||.u-v.|| by A6,A1,Lm8;
    hence ||. g/.t - h/.t .|| <= ((r*(b-a))|^(m+1) )/((m+1)!) * ||.u-v.||
      by A7,XXREAL_0:2;
   end;
   hence thesis by ORDEQ_01:27,A2,A3;
end;
