let f1, f2 be PartFunc of V,W; :: thesis: ( dom f1 = (- h) ++ (dom f) & ( for x being Element of V st x in (- h) ++ (dom f) holds
f1 . x = f . (x + h) ) & dom f2 = (- h) ++ (dom f) & ( for x being Element of V st x in (- h) ++ (dom f) holds
f2 . x = f . (x + h) ) implies f1 = f2 )
set X = (- h) ++ (dom f);
assume that
A6: dom f1 = (- h) ++ (dom f) and
A7: for x being Element of V st x in (- h) ++ (dom f) holds
f1 . x = f . (x + h) and
A8: dom f2 = (- h) ++ (dom f) and
A9: for x being Element of V st x in (- h) ++ (dom f) holds
f2 . x = f . (x + h) ; :: thesis: f1 = f2
for x being Element of V st x in dom f1 holds
f1 . x = f2 . x

proof

let x be Element of V; :: thesis: ( x in dom f1 implies f1 . x = f2 . x )
assume A10: x in dom f1 ; :: thesis: f1 . x = f2 . x
then f1 . x = f . (x + h) by A6, A7;
hence f1 . x = f2 . x by A6, A9, A10; :: thesis: verum

end;

hence f1 = f2 by A6, A8, PARTFUN1:5; :: thesis: verum