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 Th51:
  for m be Nat st a <= b & 0 < r &
  (for y1,y2 be VECTOR of REAL-NS n holds ||.G/.y1-G/.y2.|| <= r*||.y1-y2.||)
  holds
  for u,v be VECTOR of R_NormSpace_of_ContinuousFunctions([' a,b '],REAL-NS n)
   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 REAL-NS n
                               holds ||.G/.y1-G/.y2.||<=r*||.y1-y2.||;
  let u,v be VECTOR of
    R_NormSpace_of_ContinuousFunctions([' a,b '],REAL-NS n);
  reconsider u1=iter(Fredholm(G,a,b,y0),(m+1)).u as VECTOR of
    R_NormSpace_of_ContinuousFunctions([' a,b '],REAL-NS n);
  reconsider v1=iter(Fredholm(G,a,b,y0),(m+1)).v as VECTOR of
    R_NormSpace_of_ContinuousFunctions([' a,b '],REAL-NS n);
  consider g be continuous PartFunc of REAL,REAL-NS n such that
A2: u1=g & dom g = [' a,b '] by Def2;
  consider h be continuous PartFunc of REAL,REAL-NS n such that
A3: v1=h & dom h = [' a,b '] by Def2;
  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,Th50,A1,A2,A3;
    ((r*(t-a))|^(m+1) )/((m+1)!) * ||.u-v.||
    <= ((r*(b-a))|^(m+1) )/((m+1)!) * ||.u-v.|| by A6,A1,Lm13;
    hence ||. g/.t - h/.t .|| <= ((r*(b-a))|^(m+1) )/((m+1)!) * ||.u-v.||
    by A7,XXREAL_0:2;
  end;
  hence thesis by Th27,A2,A3;
end;
