let F, G be Field; :: thesis: for V being VectSp of F
for W being VectSp of G
for f1, f2 being Function of V,W
for x, h being Element of V
for n being Nat holds ((cdif ((f1 - f2),h)) . (n + 1)) /. x = (((cdif (f1,h)) . (n + 1)) /. x) - (((cdif (f2,h)) . (n + 1)) /. x)
let V be VectSp of F; :: thesis: for W being VectSp of G
for f1, f2 being Function of V,W
for x, h being Element of V
for n being Nat holds ((cdif ((f1 - f2),h)) . (n + 1)) /. x = (((cdif (f1,h)) . (n + 1)) /. x) - (((cdif (f2,h)) . (n + 1)) /. x)
let W be VectSp of G; :: thesis: for f1, f2 being Function of V,W
for x, h being Element of V
for n being Nat holds ((cdif ((f1 - f2),h)) . (n + 1)) /. x = (((cdif (f1,h)) . (n + 1)) /. x) - (((cdif (f2,h)) . (n + 1)) /. x)
let f1, f2 be Function of V,W; :: thesis: for x, h being Element of V
for n being Nat holds ((cdif ((f1 - f2),h)) . (n + 1)) /. x = (((cdif (f1,h)) . (n + 1)) /. x) - (((cdif (f2,h)) . (n + 1)) /. x)
let x, h be Element of V; :: thesis: for n being Nat holds ((cdif ((f1 - f2),h)) . (n + 1)) /. x = (((cdif (f1,h)) . (n + 1)) /. x) - (((cdif (f2,h)) . (n + 1)) /. x)
let n be Nat; :: thesis: ((cdif ((f1 - f2),h)) . (n + 1)) /. x = (((cdif (f1,h)) . (n + 1)) /. x) - (((cdif (f2,h)) . (n + 1)) /. x)
defpred S₁[ Nat] means for x being Element of V holds ((cdif ((f1 - f2),h)) . ($1 + 1)) /. x = (((cdif (f1,h)) . ($1 + 1)) /. x) - (((cdif (f2,h)) . ($1 + 1)) /. x);
A1: S₁[ 0 ]

proof

let x be Element of V; :: thesis: ((cdif ((f1 - f2),h)) . (0 + 1)) /. x = (((cdif (f1,h)) . (0 + 1)) /. x) - (((cdif (f2,h)) . (0 + 1)) /. x)
x - (((2 * (1. F)) ") * h) in the carrier of V ;
then ( x - (((2 * (1. F)) ") * h) in dom f1 & x - (((2 * (1. F)) ") * h) in dom f2 ) by FUNCT_2:def 1;
then x - (((2 * (1. F)) ") * h) in (dom f1) /\ (dom f2) by XBOOLE_0:def 4;
then A2: x - (((2 * (1. F)) ") * h) in dom (f1 - f2) by VFUNCT_1:def 2;
x + (((2 * (1. F)) ") * h) in the carrier of V ;
then ( x + (((2 * (1. F)) ") * h) in dom f1 & x + (((2 * (1. F)) ") * h) in dom f2 ) by FUNCT_2:def 1;
then x + (((2 * (1. F)) ") * h) in (dom f1) /\ (dom f2) by XBOOLE_0:def 4;
then A3: x + (((2 * (1. F)) ") * h) in dom (f1 - f2) by VFUNCT_1:def 2;
((cdif ((f1 - f2),h)) . (0 + 1)) /. x = (cD (((cdif ((f1 - f2),h)) . 0),h)) /. x by Def8
.= (cD ((f1 - f2),h)) /. x by Def8
.= ((f1 - f2) /. (x + (((2 * (1. F)) ") * h))) - ((f1 - f2) /. (x - (((2 * (1. F)) ") * h))) by Th5
.= ((f1 /. (x + (((2 * (1. F)) ") * h))) - (f2 /. (x + (((2 * (1. F)) ") * h)))) - ((f1 - f2) /. (x - (((2 * (1. F)) ") * h))) by A3, VFUNCT_1:def 2
.= ((f1 /. (x + (((2 * (1. F)) ") * h))) - (f2 /. (x + (((2 * (1. F)) ") * h)))) - ((f1 /. (x - (((2 * (1. F)) ") * h))) - (f2 /. (x - (((2 * (1. F)) ") * h)))) by A2, VFUNCT_1:def 2
.= (((f1 /. (x + (((2 * (1. F)) ") * h))) - (f2 /. (x + (((2 * (1. F)) ") * h)))) - (f1 /. (x - (((2 * (1. F)) ") * h)))) + (f2 /. (x - (((2 * (1. F)) ") * h))) by RLVECT_1:29
.= ((f1 /. (x + (((2 * (1. F)) ") * h))) - ((f1 /. (x - (((2 * (1. F)) ") * h))) + (f2 /. (x + (((2 * (1. F)) ") * h))))) + (f2 /. (x - (((2 * (1. F)) ") * h))) by RLVECT_1:27
.= (f1 /. (x + (((2 * (1. F)) ") * h))) - (((f1 /. (x - (((2 * (1. F)) ") * h))) + (f2 /. (x + (((2 * (1. F)) ") * h)))) - (f2 /. (x - (((2 * (1. F)) ") * h)))) by RLVECT_1:29
.= (f1 /. (x + (((2 * (1. F)) ") * h))) - ((f1 /. (x - (((2 * (1. F)) ") * h))) + ((f2 /. (x + (((2 * (1. F)) ") * h))) - (f2 /. (x - (((2 * (1. F)) ") * h))))) by RLVECT_1:28
.= ((f1 /. (x + (((2 * (1. F)) ") * h))) - (f1 /. (x - (((2 * (1. F)) ") * h)))) - ((f2 /. (x + (((2 * (1. F)) ") * h))) - (f2 /. (x - (((2 * (1. F)) ") * h)))) by RLVECT_1:27
.= ((cD (f1,h)) /. x) - ((f2 /. (x + (((2 * (1. F)) ") * h))) - (f2 /. (x - (((2 * (1. F)) ") * h)))) by Th5
.= ((cD (f1,h)) /. x) - ((cD (f2,h)) /. x) by Th5
.= ((cD (((cdif (f1,h)) . 0),h)) /. x) - ((cD (f2,h)) /. x) by Def8
.= ((cD (((cdif (f1,h)) . 0),h)) /. x) - ((cD (((cdif (f2,h)) . 0),h)) /. x) by Def8
.= (((cdif (f1,h)) . (0 + 1)) /. x) - ((cD (((cdif (f2,h)) . 0),h)) /. x) by Def8
.= (((cdif (f1,h)) . (0 + 1)) /. x) - (((cdif (f2,h)) . (0 + 1)) /. x) by Def8 ;
hence ((cdif ((f1 - f2),h)) . (0 + 1)) /. x = (((cdif (f1,h)) . (0 + 1)) /. x) - (((cdif (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 Element of V holds ((cdif ((f1 - f2),h)) . (k + 1)) /. x = (((cdif (f1,h)) . (k + 1)) /. x) - (((cdif (f2,h)) . (k + 1)) /. x) ; :: thesis: S₁[k + 1]
let x be Element of V; :: thesis: ((cdif ((f1 - f2),h)) . ((k + 1) + 1)) /. x = (((cdif (f1,h)) . ((k + 1) + 1)) /. x) - (((cdif (f2,h)) . ((k + 1) + 1)) /. x)
A6: ( ((cdif ((f1 - f2),h)) . (k + 1)) /. (x - (((2 * (1. F)) ") * h)) = (((cdif (f1,h)) . (k + 1)) /. (x - (((2 * (1. F)) ") * h))) - (((cdif (f2,h)) . (k + 1)) /. (x - (((2 * (1. F)) ") * h))) & ((cdif ((f1 - f2),h)) . (k + 1)) /. (x + (((2 * (1. F)) ") * h)) = (((cdif (f1,h)) . (k + 1)) /. (x + (((2 * (1. F)) ") * h))) - (((cdif (f2,h)) . (k + 1)) /. (x + (((2 * (1. F)) ") * h))) ) by A5;
A7: (cdif ((f1 - f2),h)) . (k + 1) is Function of the carrier of V, the carrier of W by Th19;
A8: (cdif (f2,h)) . (k + 1) is Function of the carrier of V, the carrier of W by Th19;
A9: (cdif (f1,h)) . (k + 1) is Function of the carrier of V, the carrier of W by Th19;
((cdif ((f1 - f2),h)) . ((k + 1) + 1)) /. x = (cD (((cdif ((f1 - f2),h)) . (k + 1)),h)) /. x by Def8
.= (((cdif ((f1 - f2),h)) . (k + 1)) /. (x + (((2 * (1. F)) ") * h))) - (((cdif ((f1 - f2),h)) . (k + 1)) /. (x - (((2 * (1. F)) ") * h))) by A7, Th5
.= (((((cdif (f1,h)) . (k + 1)) /. (x + (((2 * (1. F)) ") * h))) - (((cdif (f2,h)) . (k + 1)) /. (x + (((2 * (1. F)) ") * h)))) - (((cdif (f1,h)) . (k + 1)) /. (x - (((2 * (1. F)) ") * h)))) + (((cdif (f2,h)) . (k + 1)) /. (x - (((2 * (1. F)) ") * h))) by RLVECT_1:29, A6
.= ((((cdif (f1,h)) . (k + 1)) /. (x + (((2 * (1. F)) ") * h))) - ((((cdif (f1,h)) . (k + 1)) /. (x - (((2 * (1. F)) ") * h))) + (((cdif (f2,h)) . (k + 1)) /. (x + (((2 * (1. F)) ") * h))))) + (((cdif (f2,h)) . (k + 1)) /. (x - (((2 * (1. F)) ") * h))) by RLVECT_1:27
.= (((cdif (f1,h)) . (k + 1)) /. (x + (((2 * (1. F)) ") * h))) - (((((cdif (f1,h)) . (k + 1)) /. (x - (((2 * (1. F)) ") * h))) + (((cdif (f2,h)) . (k + 1)) /. (x + (((2 * (1. F)) ") * h)))) - (((cdif (f2,h)) . (k + 1)) /. (x - (((2 * (1. F)) ") * h)))) by RLVECT_1:29
.= (((cdif (f1,h)) . (k + 1)) /. (x + (((2 * (1. F)) ") * h))) - ((((cdif (f1,h)) . (k + 1)) /. (x - (((2 * (1. F)) ") * h))) + ((((cdif (f2,h)) . (k + 1)) /. (x + (((2 * (1. F)) ") * h))) - (((cdif (f2,h)) . (k + 1)) /. (x - (((2 * (1. F)) ") * h))))) by RLVECT_1:28
.= ((((cdif (f1,h)) . (k + 1)) /. (x + (((2 * (1. F)) ") * h))) - (((cdif (f1,h)) . (k + 1)) /. (x - (((2 * (1. F)) ") * h)))) - ((((cdif (f2,h)) . (k + 1)) /. (x + (((2 * (1. F)) ") * h))) - (((cdif (f2,h)) . (k + 1)) /. (x - (((2 * (1. F)) ") * h)))) by RLVECT_1:27
.= ((cD (((cdif (f1,h)) . (k + 1)),h)) /. x) - ((((cdif (f2,h)) . (k + 1)) /. (x + (((2 * (1. F)) ") * h))) - (((cdif (f2,h)) . (k + 1)) /. (x - (((2 * (1. F)) ") * h)))) by A9, Th5
.= ((cD (((cdif (f1,h)) . (k + 1)),h)) /. x) - ((cD (((cdif (f2,h)) . (k + 1)),h)) /. x) by A8, Th5
.= (((cdif (f1,h)) . ((k + 1) + 1)) /. x) - ((cD (((cdif (f2,h)) . (k + 1)),h)) /. x) by Def8
.= (((cdif (f1,h)) . ((k + 1) + 1)) /. x) - (((cdif (f2,h)) . ((k + 1) + 1)) /. x) by Def8 ;
hence ((cdif ((f1 - f2),h)) . ((k + 1) + 1)) /. x = (((cdif (f1,h)) . ((k + 1) + 1)) /. x) - (((cdif (f2,h)) . ((k + 1) + 1)) /. x) ; :: thesis: verum

end;

for n being Nat holds S₁[n] from NAT_1:sch 2(A1, A4);
hence ((cdif ((f1 - f2),h)) . (n + 1)) /. x = (((cdif (f1,h)) . (n + 1)) /. x) - (((cdif (f2,h)) . (n + 1)) /. x) ; :: thesis: verum