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: for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be Nat; :: thesis: ( S1[k] implies S1[k + 1] )
assume A2: 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)
A3: ( ((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 A2;
A4: (fdif ((f1 + f2),h)) . (k + 1) is Function of REAL,REAL by Th2;
A5: (fdif (f2,h)) . (k + 1) is Function of REAL,REAL by Th2;
A6: (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 A3
.= ((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;
A7: 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)
reconsider xx = x, h = h as Element of REAL by XREAL_0:def 1;
((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 . (xx + h)) + (f2 . (xx + h))) - ((f1 + f2) . xx) by VALUED_1:1
.= ((f1 . (x + h)) + (f2 . (x + h))) - ((f1 . x) + (f2 . x)) by VALUED_1:1
.= ((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;
for n being Nat holds S1[n] from NAT_1:sch 2(A7, A1);
hence ((fdif ((f1 + f2),h)) . (n + 1)) . x = (((fdif (f1,h)) . (n + 1)) . x) + (((fdif (f2,h)) . (n + 1)) . x) ; :: thesis: verum