let F, G be Field; :: thesis: for V being VectSp of F
for W being VectSp of G
for f being Function of V,W
for x, h being Element of V
for r being Element of G
for n being Nat holds ((fdif ((r (#) f),h)) . (n + 1)) /. x = r * (((fdif (f,h)) . (n + 1)) /. x)
let V be VectSp of F; :: thesis: for W being VectSp of G
for f being Function of V,W
for x, h being Element of V
for r being Element of G
for n being Nat holds ((fdif ((r (#) f),h)) . (n + 1)) /. x = r * (((fdif (f,h)) . (n + 1)) /. x)
let W be VectSp of G; :: thesis: for f being Function of V,W
for x, h being Element of V
for r being Element of G
for n being Nat holds ((fdif ((r (#) f),h)) . (n + 1)) /. x = r * (((fdif (f,h)) . (n + 1)) /. x)
let f be Function of V,W; :: thesis: for x, h being Element of V
for r being Element of G
for n being Nat holds ((fdif ((r (#) f),h)) . (n + 1)) /. x = r * (((fdif (f,h)) . (n + 1)) /. x)
let x, h be Element of V; :: thesis: for r being Element of G
for n being Nat holds ((fdif ((r (#) f),h)) . (n + 1)) /. x = r * (((fdif (f,h)) . (n + 1)) /. x)
let r be Element of G; :: thesis: for n being Nat holds ((fdif ((r (#) f),h)) . (n + 1)) /. x = r * (((fdif (f,h)) . (n + 1)) /. x)
let n be Nat; :: thesis: ((fdif ((r (#) f),h)) . (n + 1)) /. x = r * (((fdif (f,h)) . (n + 1)) /. x)
defpred S₁[ Nat] means for x being Element of V 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 Element of V holds ((fdif ((r (#) f),h)) . (k + 1)) /. x = r * (((fdif (f,h)) . (k + 1)) /. x) ; :: thesis: S₁[k + 1]
let x be Element of V; :: 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;
reconsider rfdk = (fdif ((r (#) f),h)) . (k + 1) as Function of V,W by Th2;
reconsider fdk = (fdif (f,h)) . (k + 1) as Function of V,W by Th2;
((fdif ((r (#) f),h)) . ((k + 1) + 1)) /. x = (fD (((fdif ((r (#) f),h)) . (k + 1)),h)) /. x by Def6
.= (rfdk /. (x + h)) - (rfdk /. x) by Th3
.= r * ((fdk /. (x + h)) - (fdk /. x)) by VECTSP_1:23, A3
.= r * ((fD (fdk,h)) /. x) by 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 Element of V; :: thesis: ((fdif ((r (#) f),h)) . (0 + 1)) /. x = r * (((fdif (f,h)) . (0 + 1)) /. x)
x in the carrier of V ;
then A7: x in dom (r (#) f) by FUNCT_2:def 1;
x + h in the carrier of V ;
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, Def4X
.= (r * (f /. (x + h))) - (r * (f /. x)) by A7, Def4X
.= r * ((f /. (x + h)) - (f /. x)) by VECTSP_1:23
.= 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