let n be Nat; :: thesis: for h, r, x being Real
for f being Function of REAL,REAL holds ((fdif ((r (#) f),h)) . (n + 1)) . x = r * (((fdif (f,h)) . (n + 1)) . x)
let h, r, x be Real; :: thesis: for f being Function of REAL,REAL holds ((fdif ((r (#) f),h)) . (n + 1)) . x = r * (((fdif (f,h)) . (n + 1)) . x)
let f be Function of REAL,REAL; :: thesis: ((fdif ((r (#) f),h)) . (n + 1)) . x = r * (((fdif (f,h)) . (n + 1)) . x)
defpred S₁[ Nat] means for x being Real holds ((fdif ((r (#) f),h)) . ($1 + 1)) . x = r * (((fdif (f,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 ((r (#) f),h)) . (k + 1)) . x = r * (((fdif (f,h)) . (k + 1)) . x) ; :: thesis: S₁[k + 1]
let x be Real; :: thesis: ((fdif ((r (#) f),h)) . ((k + 1) + 1)) . x = r * (((fdif (f,h)) . ((k + 1) + 1)) . x)
A3: ( ((fdif ((r (#) f),h)) . (k + 1)) . x = r * (((fdif (f,h)) . (k + 1)) . x) & ((fdif ((r (#) f),h)) . (k + 1)) . (x + h) = r * (((fdif (f,h)) . (k + 1)) . (x + h)) ) by A2;
A4: (fdif ((r (#) f),h)) . (k + 1) is Function of REAL,REAL by Th2;
A5: (fdif (f,h)) . (k + 1) is Function of REAL,REAL by Th2;
((fdif ((r (#) f),h)) . ((k + 1) + 1)) . x = (fD (((fdif ((r (#) f),h)) . (k + 1)),h)) . x by Def6
.= (r * (((fdif (f,h)) . (k + 1)) . (x + h))) - (r * (((fdif (f,h)) . (k + 1)) . x)) by A3, A4, Th3
.= r * ((((fdif (f,h)) . (k + 1)) . (x + h)) - (((fdif (f,h)) . (k + 1)) . x))
.= r * ((fD (((fdif (f,h)) . (k + 1)),h)) . x) by A5, Th3
.= r * (((fdif (f,h)) . ((k + 1) + 1)) . x) by Def6 ;
hence ((fdif ((r (#) f),h)) . ((k + 1) + 1)) . x = r * (((fdif (f,h)) . ((k + 1) + 1)) . x) ; :: thesis: verum

end;

A6: S₁[ 0 ]

proof

let x be Real; :: thesis: ((fdif ((r (#) f),h)) . (0 + 1)) . x = r * (((fdif (f,h)) . (0 + 1)) . x)
x in REAL by XREAL_0:def 1;
then A7: x in dom (r (#) f) by FUNCT_2:def 1;
x + h in REAL by XREAL_0:def 1;
then A8: x + h in dom (r (#) f) by FUNCT_2:def 1;
((fdif ((r (#) f),h)) . (0 + 1)) . x = (fD (((fdif ((r (#) f),h)) . 0),h)) . x by Def6
.= (fD ((r (#) f),h)) . x by Def6
.= ((r (#) f) . (x + h)) - ((r (#) f) . x) by Th3
.= (r * (f . (x + h))) - ((r (#) f) . x) by A8, VALUED_1:def 5
.= (r * (f . (x + h))) - (r * (f . x)) by A7, VALUED_1:def 5
.= r * ((f . (x + h)) - (f . x))
.= r * ((fD (f,h)) . x) by Th3
.= r * ((fD (((fdif (f,h)) . 0),h)) . x) by Def6
.= r * (((fdif (f,h)) . (0 + 1)) . x) by Def6 ;
hence ((fdif ((r (#) f),h)) . (0 + 1)) . x = r * (((fdif (f,h)) . (0 + 1)) . x) ; :: thesis: verum

end;

for n being Nat holds S₁[n] from NAT_1:sch 2(A6, A1);
hence ((fdif ((r (#) f),h)) . (n + 1)) . x = r * (((fdif (f,h)) . (n + 1)) . x) ; :: thesis: verum