theorem Th55: :: ORDEQ_02:24

for X being RealBanachSpace
for a, b, r being Real
for y0 being VECTOR of X
for G being Function of X,X
for m being Nat st a <= b & 0 < r & ( for y1, y2 being VECTOR of X holds ||.((G /. y1) - (G /. y2)).|| <= r * ||.(y1 - y2).|| ) holds
for u, v being 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).||