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 Th56:
a < b & G is_Lipschitzian_on the carrier of X implies
  ex m be Nat st iter(Fredholm(G,a,b,y0),(m+1)) is contraction
proof
   assume A1: a<b & G is_Lipschitzian_on the carrier of X; then
   consider r be Real such that
A2: 0<r & for x1,x2 be Point of X
     st x1 in the carrier of X & x2 in the carrier of X holds
      ||.G/.x1-G/.x2.||<=r*||.x1-x2.||;
A3:for x1,x2 be Point of X holds ||.G/.x1-G/.x2.||<=r*||.x1-x2.|| by A2;
   consider m be Element of NAT such that
A4: ((r*(b-a))|^(m+1))/((m+1)!) < 1
  & 0 < ((r*(b-a))|^(m+1))/((m+1)!) by Lm9,A1,A2;
   take m;
   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.|| by Th55,A3,A2,A1;
   hence thesis by A4;
end;
