let n be Nat; :: thesis: for h, x being Real
for f1, f2 being Function of REAL,REAL holds ((bdif ((f1 + f2),h)) . (n + 1)) . x = (((bdif (f1,h)) . (n + 1)) . x) + (((bdif (f2,h)) . (n + 1)) . x)
let h, x be Real; :: thesis: for f1, f2 being Function of REAL,REAL holds ((bdif ((f1 + f2),h)) . (n + 1)) . x = (((bdif (f1,h)) . (n + 1)) . x) + (((bdif (f2,h)) . (n + 1)) . x)
let f1, f2 be Function of REAL,REAL; :: thesis: ((bdif ((f1 + f2),h)) . (n + 1)) . x = (((bdif (f1,h)) . (n + 1)) . x) + (((bdif (f2,h)) . (n + 1)) . x)
defpred S₁[ Nat] means for x being Real holds ((bdif ((f1 + f2),h)) . ($1 + 1)) . x = (((bdif (f1,h)) . ($1 + 1)) . x) + (((bdif (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 ((bdif ((f1 + f2),h)) . (k + 1)) . x = (((bdif (f1,h)) . (k + 1)) . x) + (((bdif (f2,h)) . (k + 1)) . x) ; :: thesis: S₁[k + 1]
let x be Real; :: thesis: ((bdif ((f1 + f2),h)) . ((k + 1) + 1)) . x = (((bdif (f1,h)) . ((k + 1) + 1)) . x) + (((bdif (f2,h)) . ((k + 1) + 1)) . x)
A3: ( ((bdif ((f1 + f2),h)) . (k + 1)) . x = (((bdif (f1,h)) . (k + 1)) . x) + (((bdif (f2,h)) . (k + 1)) . x) & ((bdif ((f1 + f2),h)) . (k + 1)) . (x - h) = (((bdif (f1,h)) . (k + 1)) . (x - h)) + (((bdif (f2,h)) . (k + 1)) . (x - h)) ) by A2;
A4: (bdif ((f1 + f2),h)) . (k + 1) is Function of REAL,REAL by Th12;
A5: (bdif (f2,h)) . (k + 1) is Function of REAL,REAL by Th12;
A6: (bdif (f1,h)) . (k + 1) is Function of REAL,REAL by Th12;
((bdif ((f1 + f2),h)) . ((k + 1) + 1)) . x = (bD (((bdif ((f1 + f2),h)) . (k + 1)),h)) . x by Def7
.= (((bdif ((f1 + f2),h)) . (k + 1)) . x) - (((bdif ((f1 + f2),h)) . (k + 1)) . (x - h)) by A4, Th4
.= ((((bdif (f1,h)) . (k + 1)) . x) - (((bdif (f1,h)) . (k + 1)) . (x - h))) + ((((bdif (f2,h)) . (k + 1)) . x) - (((bdif (f2,h)) . (k + 1)) . (x - h))) by A3
.= ((bD (((bdif (f1,h)) . (k + 1)),h)) . x) + ((((bdif (f2,h)) . (k + 1)) . x) - (((bdif (f2,h)) . (k + 1)) . (x - h))) by A6, Th4
.= ((bD (((bdif (f1,h)) . (k + 1)),h)) . x) + ((bD (((bdif (f2,h)) . (k + 1)),h)) . x) by A5, Th4
.= (((bdif (f1,h)) . ((k + 1) + 1)) . x) + ((bD (((bdif (f2,h)) . (k + 1)),h)) . x) by Def7
.= (((bdif (f1,h)) . ((k + 1) + 1)) . x) + (((bdif (f2,h)) . ((k + 1) + 1)) . x) by Def7 ;
hence ((bdif ((f1 + f2),h)) . ((k + 1) + 1)) . x = (((bdif (f1,h)) . ((k + 1) + 1)) . x) + (((bdif (f2,h)) . ((k + 1) + 1)) . x) ; :: thesis: verum

end;

A7: S₁[ 0 ]

proof

let x be Real; :: thesis: ((bdif ((f1 + f2),h)) . (0 + 1)) . x = (((bdif (f1,h)) . (0 + 1)) . x) + (((bdif (f2,h)) . (0 + 1)) . x)
reconsider xx = x, h = h as Element of REAL by XREAL_0:def 1;
((bdif ((f1 + f2),h)) . (0 + 1)) . x = (bD (((bdif ((f1 + f2),h)) . 0),h)) . x by Def7
.= (bD ((f1 + f2),h)) . x by Def7
.= ((f1 + f2) . x) - ((f1 + f2) . (x - h)) by Th4
.= ((f1 . xx) + (f2 . xx)) - ((f1 + f2) . (xx - h)) by VALUED_1:1
.= ((f1 . x) + (f2 . x)) - ((f1 . (x - h)) + (f2 . (x - h))) by VALUED_1:1
.= ((f1 . x) - (f1 . (x - h))) + ((f2 . x) - (f2 . (x - h)))
.= ((bD (f1,h)) . x) + ((f2 . x) - (f2 . (x - h))) by Th4
.= ((bD (f1,h)) . x) + ((bD (f2,h)) . x) by Th4
.= ((bD (((bdif (f1,h)) . 0),h)) . x) + ((bD (f2,h)) . x) by Def7
.= ((bD (((bdif (f1,h)) . 0),h)) . x) + ((bD (((bdif (f2,h)) . 0),h)) . x) by Def7
.= (((bdif (f1,h)) . (0 + 1)) . x) + ((bD (((bdif (f2,h)) . 0),h)) . x) by Def7
.= (((bdif (f1,h)) . (0 + 1)) . x) + (((bdif (f2,h)) . (0 + 1)) . x) by Def7 ;
hence ((bdif ((f1 + f2),h)) . (0 + 1)) . x = (((bdif (f1,h)) . (0 + 1)) . x) + (((bdif (f2,h)) . (0 + 1)) . x) ; :: thesis: verum

end;

for n being Nat holds S₁[n] from NAT_1:sch 2(A7, A1);
hence ((bdif ((f1 + f2),h)) . (n + 1)) . x = (((bdif (f1,h)) . (n + 1)) . x) + (((bdif (f2,h)) . (n + 1)) . x) ; :: thesis: verum