let f1, f2 be PartFunc of REAL,REAL; :: thesis: ( dom f1 = (- h) ++ (dom f) & ( for x being Real st x in (- h) ++ (dom f) holds
f1 . x = f . (x + h) ) & dom f2 = (- h) ++ (dom f) & ( for x being Real st x in (- h) ++ (dom f) holds
f2 . x = f . (x + h) ) implies f1 = f2 )
set X = (- h) ++ (dom f);
assume that
A3: dom f1 = (- h) ++ (dom f) and
A4: for x being Real st x in (- h) ++ (dom f) holds
f1 . x = f . (x + h) and
A5: dom f2 = (- h) ++ (dom f) and
A6: for x being Real st x in (- h) ++ (dom f) holds
f2 . x = f . (x + h) ; :: thesis: f1 = f2
for x being Element of REAL st x in dom f1 holds
f1 . x = f2 . x

proof

let x be Element of REAL ; :: thesis: ( x in dom f1 implies f1 . x = f2 . x )
assume A7: x in dom f1 ; :: thesis: f1 . x = f2 . x
then f1 . x = f . (x + h) by A3, A4;
hence f1 . x = f2 . x by A3, A6, A7; :: thesis: verum

end;

hence f1 = f2 by A3, A5, PARTFUN1:5; :: thesis: verum