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
for y1,y2 be continuous PartFunc of REAL,the carrier of X
   st a < b & Z = ].a,b.[ & G is_Lipschitzian_on the carrier of X
    & dom y1 = ['a,b'] & y1 is_differentiable_on Z & y1/.a = y0
    & (for t be Real st t in Z holds diff(y1,t) = G.(y1/.t))
    & dom y2 = ['a,b'] & y2 is_differentiable_on Z & y2/.a = y0
    & (for t be Real st t in Z holds diff(y2,t) = G.(y2/.t))
   holds y1=y2
proof
   let y1,y2 be continuous PartFunc of REAL,the carrier of X;
   assume
A1:a < b & Z = ].a,b.[ & G is_Lipschitzian_on the carrier of X
   & dom y1 = ['a,b'] & y1 is_differentiable_on Z & y1/.a = y0
   & (for t be Real st t in Z holds diff(y1,t) = G.(y1/.t))
   & dom y2 = ['a,b'] & y2 is_differentiable_on Z & y2/.a = y0
   & (for t be Real st t in Z holds diff(y2,t) = G.(y2/.t)); then
   Fredholm(G,a,b,y0) is with_unique_fixpoint by Th57; then
   consider y being set such that
SS: y is_a_fixpoint_of Fredholm(G,a,b,y0)
  & for z being set st z is_a_fixpoint_of Fredholm(G,a,b,y0) holds y = z;
   y1 = y by Th59,A1,SS .= y2 by Th59,A1,SS;
   hence thesis;
end;
