let h, x be Real; :: thesis: for f being PartFunc of REAL,REAL st x in dom f & x - h in dom f holds

(bD (f,h)) . x = (f . x) - (f . (x - h))

let f be PartFunc of REAL,REAL; :: thesis: ( x in dom f & x - h in dom f implies (bD (f,h)) . x = (f . x) - (f . (x - h)) )

assume A1: ( x in dom f & x - h in dom f ) ; :: thesis: (bD (f,h)) . x = (f . x) - (f . (x - h))

A2: dom (Shift (f,(- h))) = (- (- h)) ++ (dom f) by Def1;

A3: h + (x - (- (- h))) in (- (- h)) ++ (dom f) by A1, MEASURE6:46;

then A4: (Shift (f,(- h))) . x = f . (x + (- h)) by Def1;

x in (dom (Shift (f,(- h)))) /\ (dom f) by A3, A2, A1, XBOOLE_0:def 4;

then x in dom (bD (f,h)) by VALUED_1:12;

hence (bD (f,h)) . x = (f . x) - (f . (x - h)) by A4, VALUED_1:13; :: thesis: verum

(bD (f,h)) . x = (f . x) - (f . (x - h))

let f be PartFunc of REAL,REAL; :: thesis: ( x in dom f & x - h in dom f implies (bD (f,h)) . x = (f . x) - (f . (x - h)) )

assume A1: ( x in dom f & x - h in dom f ) ; :: thesis: (bD (f,h)) . x = (f . x) - (f . (x - h))

A2: dom (Shift (f,(- h))) = (- (- h)) ++ (dom f) by Def1;

A3: h + (x - (- (- h))) in (- (- h)) ++ (dom f) by A1, MEASURE6:46;

then A4: (Shift (f,(- h))) . x = f . (x + (- h)) by Def1;

x in (dom (Shift (f,(- h)))) /\ (dom f) by A3, A2, A1, XBOOLE_0:def 4;

then x in dom (bD (f,h)) by VALUED_1:12;

hence (bD (f,h)) . x = (f . x) - (f . (x - h)) by A4, VALUED_1:13; :: thesis: verum