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 Th57:
  a < b & G is_Lipschitzian_on the carrier of X implies
    Fredholm(G,a,b,y0) is with_unique_fixpoint
proof
   assume a<b & G is_Lipschitzian_on the carrier of X; then
   ex m be Nat st iter(Fredholm(G,a,b,y0),m+1) is contraction by Th56;
   hence thesis by ORDEQ_01:7;
end;
