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 Th52:
  a<b & G is_Lipschitzian_on the carrier of REAL-NS n 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 REAL-NS n; then
  consider r be Real such that
A2: 0<r & for x1,x2 be Point of REAL-NS n
     st x1 in (the carrier of REAL-NS n)
      & x2 in (the carrier of REAL-NS n) holds
   ||.G/.x1-G/.x2.||<=r*||.x1-x2.|| by NFCONT_1:def 9;
A3: for x1,x2 be Point of REAL-NS n 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 Lm14,A1,A2;
  take m;
  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.|| by Th51,A3,A2,A1;
  hence thesis by A4;
end;
