let n be Element of NAT ; :: thesis: for h, x being Real
for f1, f2 being Function of REAL ,REAL holds ((cdif (f1 - f2),h) . (n + 1)) . x = (((cdif f1,h) . (n + 1)) . x) - (((cdif f2,h) . (n + 1)) . x)
let h, x be Real; :: thesis: for f1, f2 being Function of REAL ,REAL 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 REAL ,REAL ; :: thesis: ((cdif (f1 - f2),h) . (n + 1)) . x = (((cdif f1,h) . (n + 1)) . x) - (((cdif f2,h) . (n + 1)) . x)
defpred S₁[ Element of NAT ] means for x being Real 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 Real; :: thesis: ((cdif (f1 - f2),h) . (0 + 1)) . x = (((cdif f1,h) . (0 + 1)) . x) - (((cdif f2,h) . (0 + 1)) . x)
x - (h / 2) in REAL ;
then ( x - (h / 2) in dom f1 & x - (h / 2) in dom f2 ) by FUNCT_2:def 1;
then x - (h / 2) in (dom f1) /\ (dom f2) by XBOOLE_0:def 4;
then A2: x - (h / 2) in dom (f1 - f2) by VALUED_1:12;
x + (h / 2) in REAL ;
then ( x + (h / 2) in dom f1 & x + (h / 2) in dom f2 ) by FUNCT_2:def 1;
then x + (h / 2) in (dom f1) /\ (dom f2) by XBOOLE_0:def 4;
then A3: x + (h / 2) in dom (f1 - f2) by VALUED_1:12;
((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 + (h / 2))) - ((f1 - f2) . (x - (h / 2))) by Th5
.= ((f1 . (x + (h / 2))) - (f2 . (x + (h / 2)))) - ((f1 - f2) . (x - (h / 2))) by A3, VALUED_1:13
.= ((f1 . (x + (h / 2))) - (f2 . (x + (h / 2)))) - ((f1 . (x - (h / 2))) - (f2 . (x - (h / 2)))) by A2, VALUED_1:13
.= ((f1 . (x + (h / 2))) - (f1 . (x - (h / 2)))) - ((f2 . (x + (h / 2))) - (f2 . (x - (h / 2))))
.= ((cD f1,h) . x) - ((f2 . (x + (h / 2))) - (f2 . (x - (h / 2)))) 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 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 A5: for x being Real 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 Real; :: 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 - (h / 2)) = (((cdif f1,h) . (k + 1)) . (x - (h / 2))) - (((cdif f2,h) . (k + 1)) . (x - (h / 2))) & ((cdif (f1 - f2),h) . (k + 1)) . (x + (h / 2)) = (((cdif f1,h) . (k + 1)) . (x + (h / 2))) - (((cdif f2,h) . (k + 1)) . (x + (h / 2))) ) by A5;
A7: (cdif (f1 - f2),h) . (k + 1) is Function of REAL ,REAL by Th19;
A8: (cdif f2,h) . (k + 1) is Function of REAL ,REAL by Th19;
A9: (cdif f1,h) . (k + 1) is Function of REAL ,REAL 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 + (h / 2))) - (((cdif (f1 - f2),h) . (k + 1)) . (x - (h / 2))) by A7, Th5
.= ((((cdif f1,h) . (k + 1)) . (x + (h / 2))) - (((cdif f1,h) . (k + 1)) . (x - (h / 2)))) - ((((cdif f2,h) . (k + 1)) . (x + (h / 2))) - (((cdif f2,h) . (k + 1)) . (x - (h / 2)))) by A6
.= ((cD ((cdif f1,h) . (k + 1)),h) . x) - ((((cdif f2,h) . (k + 1)) . (x + (h / 2))) - (((cdif f2,h) . (k + 1)) . (x - (h / 2)))) 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 Element of NAT holds S₁[n] from NAT_1:sch 1(A1, A4);
hence ((cdif (f1 - f2),h) . (n + 1)) . x = (((cdif f1,h) . (n + 1)) . x) - (((cdif f2,h) . (n + 1)) . x) ; :: thesis: verum