let n be Element of NAT ; :: thesis: for r, h, x being Real
for f being Function of REAL ,REAL holds ((cdif (r (#) f),h) . (n + 1)) . x = r * (((cdif f,h) . (n + 1)) . x)
let r, h, x be Real; :: thesis: for f being Function of REAL ,REAL holds ((cdif (r (#) f),h) . (n + 1)) . x = r * (((cdif f,h) . (n + 1)) . x)
let f be Function of REAL ,REAL ; :: thesis: ((cdif (r (#) f),h) . (n + 1)) . x = r * (((cdif f,h) . (n + 1)) . x)
defpred S₁[ Element of NAT ] means for x being Real holds ((cdif (r (#) f),h) . ($1 + 1)) . x = r * (((cdif f,h) . ($1 + 1)) . x);
A1: for k being Element of NAT st S₁[k] holds
S₁[k + 1]

proof

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A2: for x being Real holds ((cdif (r (#) f),h) . (k + 1)) . x = r * (((cdif f,h) . (k + 1)) . x) ; :: thesis: S₁[k + 1]
let x be Real; :: thesis: ((cdif (r (#) f),h) . ((k + 1) + 1)) . x = r * (((cdif f,h) . ((k + 1) + 1)) . x)
A3: ( ((cdif (r (#) f),h) . (k + 1)) . (x - (h / 2)) = r * (((cdif f,h) . (k + 1)) . (x - (h / 2))) & ((cdif (r (#) f),h) . (k + 1)) . (x + (h / 2)) = r * (((cdif f,h) . (k + 1)) . (x + (h / 2))) ) by A2;
A4: (cdif (r (#) f),h) . (k + 1) is Function of REAL ,REAL by Th19;
A5: (cdif f,h) . (k + 1) is Function of REAL ,REAL by Th19;
((cdif (r (#) f),h) . ((k + 1) + 1)) . x = (cD ((cdif (r (#) f),h) . (k + 1)),h) . x by Def8
.= (((cdif (r (#) f),h) . (k + 1)) . (x + (h / 2))) - (((cdif (r (#) f),h) . (k + 1)) . (x - (h / 2))) by A4, Th5
.= r * ((((cdif f,h) . (k + 1)) . (x + (h / 2))) - (((cdif f,h) . (k + 1)) . (x - (h / 2)))) by A3
.= r * ((cD ((cdif f,h) . (k + 1)),h) . x) by A5, Th5
.= r * (((cdif f,h) . ((k + 1) + 1)) . x) by Def8 ;
hence ((cdif (r (#) f),h) . ((k + 1) + 1)) . x = r * (((cdif f,h) . ((k + 1) + 1)) . x) ; :: thesis: verum

end;

A6: S₁[ 0 ]

proof

let x be Real; :: thesis: ((cdif (r (#) f),h) . (0 + 1)) . x = r * (((cdif f,h) . (0 + 1)) . x)
x + (h / 2) in REAL ;
then A7: x + (h / 2) in dom (r (#) f) by FUNCT_2:def 1;
x - (h / 2) in REAL ;
then A8: x - (h / 2) in dom (r (#) f) by FUNCT_2:def 1;
((cdif (r (#) f),h) . (0 + 1)) . x = (cD ((cdif (r (#) f),h) . 0 ),h) . x by Def8
.= (cD (r (#) f),h) . x by Def8
.= ((r (#) f) . (x + (h / 2))) - ((r (#) f) . (x - (h / 2))) by Th5
.= (r * (f . (x + (h / 2)))) - ((r (#) f) . (x - (h / 2))) by A7, VALUED_1:def 5
.= (r * (f . (x + (h / 2)))) - (r * (f . (x - (h / 2)))) by A8, VALUED_1:def 5
.= r * ((f . (x + (h / 2))) - (f . (x - (h / 2))))
.= r * ((cD f,h) . x) by Th5
.= r * ((cD ((cdif f,h) . 0 ),h) . x) by Def8
.= r * (((cdif f,h) . (0 + 1)) . x) by Def8 ;
hence ((cdif (r (#) f),h) . (0 + 1)) . x = r * (((cdif f,h) . (0 + 1)) . x) ; :: thesis: verum

end;

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