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
  for y1,y2 be continuous PartFunc of REAL,REAL-NS n
    st a<b & Z = ]. a,b .[ & G is_Lipschitzian_on the carrier of REAL-NS n
     & 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,REAL-NS n;
  assume A1:
     a<b & Z = ]. a,b .[ & G is_Lipschitzian_on the carrier of REAL-NS n
   & 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 Th53; then
   consider y being set such that
A2: 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;
A3: y1 is_a_fixpoint_of (Fredholm(G,a,b,y0)) by Th55,A1;
A4: y2 is_a_fixpoint_of (Fredholm(G,a,b,y0)) by Th55,A1;
  y1 = y by A3,A2 .= y2 by A4,A2;
  hence thesis;
end;
