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

end;

A6: S₁[ 0 ]

proof

let x be Real; :: thesis: ((bdif ((r (#) f),h)) . (0 + 1)) . x = r * (((bdif (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;
((bdif ((r (#) f),h)) . (0 + 1)) . x = (bD (((bdif ((r (#) f),h)) . 0),h)) . x by Def7
.= (bD ((r (#) f),h)) . x by Def7
.= ((r (#) f) . x) - ((r (#) f) . (x - h)) by Th4
.= ((r (#) f) . x) - (r * (f . (x - h))) by A8, VALUED_1:def 5
.= (r * (f . x)) - (r * (f . (x - h))) by A7, VALUED_1:def 5
.= r * ((f . x) - (f . (x - h)))
.= r * ((bD (f,h)) . x) by Th4
.= r * ((bD (((bdif (f,h)) . 0),h)) . x) by Def7
.= r * (((bdif (f,h)) . (0 + 1)) . x) by Def7 ;
hence ((bdif ((r (#) f),h)) . (0 + 1)) . x = r * (((bdif (f,h)) . (0 + 1)) . x) ; :: thesis: verum

end;

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