let n be Nat; :: thesis: for h, x being Real
for f1, f2 being Function of REAL,REAL holds ((fdif ((f1 - f2),h)) . (n + 1)) . x = (((fdif (f1,h)) . (n + 1)) . x) - (((fdif (f2,h)) . (n + 1)) . x)

let h, x be Real; :: thesis: for f1, f2 being Function of REAL,REAL holds ((fdif ((f1 - f2),h)) . (n + 1)) . x = (((fdif (f1,h)) . (n + 1)) . x) - (((fdif (f2,h)) . (n + 1)) . x)
let f1, f2 be Function of REAL,REAL; :: thesis: ((fdif ((f1 - f2),h)) . (n + 1)) . x = (((fdif (f1,h)) . (n + 1)) . x) - (((fdif (f2,h)) . (n + 1)) . x)
defpred S1[ Nat] means for x being Real holds ((fdif ((f1 - f2),h)) . (\$1 + 1)) . x = (((fdif (f1,h)) . (\$1 + 1)) . x) - (((fdif (f2,h)) . (\$1 + 1)) . x);
A1: S1[ 0 ]
proof
let x be Real; :: thesis: ((fdif ((f1 - f2),h)) . (0 + 1)) . x = (((fdif (f1,h)) . (0 + 1)) . x) - (((fdif (f2,h)) . (0 + 1)) . x)
x in REAL by XREAL_0:def 1;
then ( x in dom f1 & x in dom f2 ) by FUNCT_2:def 1;
then x in (dom f1) /\ (dom f2) by XBOOLE_0:def 4;
then A2: x in dom (f1 - f2) by VALUED_1:12;
x + h in REAL by XREAL_0:def 1;
then ( x + h in dom f1 & x + h in dom f2 ) by FUNCT_2:def 1;
then x + h in (dom f1) /\ (dom f2) by XBOOLE_0:def 4;
then A3: x + h in dom (f1 - f2) by VALUED_1:12;
((fdif ((f1 - f2),h)) . (0 + 1)) . x = (fD (((fdif ((f1 - f2),h)) . 0),h)) . x by Def6
.= (fD ((f1 - f2),h)) . x by Def6
.= ((f1 - f2) . (x + h)) - ((f1 - f2) . x) by Th3
.= ((f1 . (x + h)) - (f2 . (x + h))) - ((f1 - f2) . x) by
.= ((f1 . (x + h)) - (f2 . (x + h))) - ((f1 . x) - (f2 . x)) by
.= ((f1 . (x + h)) - (f1 . x)) - ((f2 . (x + h)) - (f2 . x))
.= ((fD (f1,h)) . x) - ((f2 . (x + h)) - (f2 . x)) by Th3
.= ((fD (f1,h)) . x) - ((fD (f2,h)) . x) by Th3
.= ((fD (((fdif (f1,h)) . 0),h)) . x) - ((fD (f2,h)) . x) by Def6
.= ((fD (((fdif (f1,h)) . 0),h)) . x) - ((fD (((fdif (f2,h)) . 0),h)) . x) by Def6
.= (((fdif (f1,h)) . (0 + 1)) . x) - ((fD (((fdif (f2,h)) . 0),h)) . x) by Def6
.= (((fdif (f1,h)) . (0 + 1)) . x) - (((fdif (f2,h)) . (0 + 1)) . x) by Def6 ;
hence ((fdif ((f1 - f2),h)) . (0 + 1)) . x = (((fdif (f1,h)) . (0 + 1)) . x) - (((fdif (f2,h)) . (0 + 1)) . x) ; :: thesis: verum
end;
A4: for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be Nat; :: thesis: ( S1[k] implies S1[k + 1] )
assume A5: for x being Real holds ((fdif ((f1 - f2),h)) . (k + 1)) . x = (((fdif (f1,h)) . (k + 1)) . x) - (((fdif (f2,h)) . (k + 1)) . x) ; :: thesis: S1[k + 1]
let x be Real; :: thesis: ((fdif ((f1 - f2),h)) . ((k + 1) + 1)) . x = (((fdif (f1,h)) . ((k + 1) + 1)) . x) - (((fdif (f2,h)) . ((k + 1) + 1)) . x)
A6: ( ((fdif ((f1 - f2),h)) . (k + 1)) . x = (((fdif (f1,h)) . (k + 1)) . x) - (((fdif (f2,h)) . (k + 1)) . x) & ((fdif ((f1 - f2),h)) . (k + 1)) . (x + h) = (((fdif (f1,h)) . (k + 1)) . (x + h)) - (((fdif (f2,h)) . (k + 1)) . (x + h)) ) by A5;
A7: (fdif ((f1 - f2),h)) . (k + 1) is Function of REAL,REAL by Th2;
A8: (fdif (f2,h)) . (k + 1) is Function of REAL,REAL by Th2;
A9: (fdif (f1,h)) . (k + 1) is Function of REAL,REAL by Th2;
((fdif ((f1 - f2),h)) . ((k + 1) + 1)) . x = (fD (((fdif ((f1 - f2),h)) . (k + 1)),h)) . x by Def6
.= (((fdif ((f1 - f2),h)) . (k + 1)) . (x + h)) - (((fdif ((f1 - f2),h)) . (k + 1)) . x) by
.= ((((fdif (f1,h)) . (k + 1)) . (x + h)) - (((fdif (f1,h)) . (k + 1)) . x)) - ((((fdif (f2,h)) . (k + 1)) . (x + h)) - (((fdif (f2,h)) . (k + 1)) . x)) by A6
.= ((fD (((fdif (f1,h)) . (k + 1)),h)) . x) - ((((fdif (f2,h)) . (k + 1)) . (x + h)) - (((fdif (f2,h)) . (k + 1)) . x)) by
.= ((fD (((fdif (f1,h)) . (k + 1)),h)) . x) - ((fD (((fdif (f2,h)) . (k + 1)),h)) . x) by
.= (((fdif (f1,h)) . ((k + 1) + 1)) . x) - ((fD (((fdif (f2,h)) . (k + 1)),h)) . x) by Def6
.= (((fdif (f1,h)) . ((k + 1) + 1)) . x) - (((fdif (f2,h)) . ((k + 1) + 1)) . x) by Def6 ;
hence ((fdif ((f1 - f2),h)) . ((k + 1) + 1)) . x = (((fdif (f1,h)) . ((k + 1) + 1)) . x) - (((fdif (f2,h)) . ((k + 1) + 1)) . x) ; :: thesis: verum
end;
for n being Nat holds S1[n] from NAT_1:sch 2(A1, A4);
hence ((fdif ((f1 - f2),h)) . (n + 1)) . x = (((fdif (f1,h)) . (n + 1)) . x) - (((fdif (f2,h)) . (n + 1)) . x) ; :: thesis: verum