let n be Nat; :: thesis: for h, x being Real
for f1, f2 being Function of REAL,REAL holds ((bdif ((f1 - f2),h)) . (n + 1)) . x = (((bdif (f1,h)) . (n + 1)) . x) - (((bdif (f2,h)) . (n + 1)) . x)
let h, x be Real; :: thesis: for f1, f2 being Function of REAL,REAL holds ((bdif ((f1 - f2),h)) . (n + 1)) . x = (((bdif (f1,h)) . (n + 1)) . x) - (((bdif (f2,h)) . (n + 1)) . x)
let f1, f2 be Function of REAL,REAL; :: thesis: ((bdif ((f1 - f2),h)) . (n + 1)) . x = (((bdif (f1,h)) . (n + 1)) . x) - (((bdif (f2,h)) . (n + 1)) . x)
defpred S₁[ Nat] means for x being Real holds ((bdif ((f1 - f2),h)) . ($1 + 1)) . x = (((bdif (f1,h)) . ($1 + 1)) . x) - (((bdif (f2,h)) . ($1 + 1)) . x);
A1: S₁[ 0 ]

proof

let x be Real; :: thesis: ((bdif ((f1 - f2),h)) . (0 + 1)) . x = (((bdif (f1,h)) . (0 + 1)) . x) - (((bdif (f2,h)) . (0 + 1)) . x)
x in REAL by XREAL_0:def 1;
then ( x in dom f1 & x in dom f2 ) by FUNCT_2:def 1;
then x in (dom f1) /\ (dom f2) by XBOOLE_0:def 4;
then A2: x in dom (f1 - f2) by VALUED_1:12;
x - h in REAL by XREAL_0:def 1;
then ( x - h in dom f1 & x - h in dom f2 ) by FUNCT_2:def 1;
then x - h in (dom f1) /\ (dom f2) by XBOOLE_0:def 4;
then A3: x - h in dom (f1 - f2) by VALUED_1:12;
((bdif ((f1 - f2),h)) . (0 + 1)) . x = (bD (((bdif ((f1 - f2),h)) . 0),h)) . x by Def7
.= (bD ((f1 - f2),h)) . x by Def7
.= ((f1 - f2) . x) - ((f1 - f2) . (x - h)) by Th4
.= ((f1 . x) - (f2 . x)) - ((f1 - f2) . (x - h)) by A2, VALUED_1:13
.= ((f1 . x) - (f2 . x)) - ((f1 . (x - h)) - (f2 . (x - h))) by A3, VALUED_1:13
.= ((f1 . x) - (f1 . (x - h))) - ((f2 . x) - (f2 . (x - h)))
.= ((bD (f1,h)) . x) - ((f2 . x) - (f2 . (x - h))) by Th4
.= ((bD (f1,h)) . x) - ((bD (f2,h)) . x) by Th4
.= ((bD (((bdif (f1,h)) . 0),h)) . x) - ((bD (f2,h)) . x) by Def7
.= ((bD (((bdif (f1,h)) . 0),h)) . x) - ((bD (((bdif (f2,h)) . 0),h)) . x) by Def7
.= (((bdif (f1,h)) . (0 + 1)) . x) - ((bD (((bdif (f2,h)) . 0),h)) . x) by Def7
.= (((bdif (f1,h)) . (0 + 1)) . x) - (((bdif (f2,h)) . (0 + 1)) . x) by Def7 ;
hence ((bdif ((f1 - f2),h)) . (0 + 1)) . x = (((bdif (f1,h)) . (0 + 1)) . x) - (((bdif (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 Real holds ((bdif ((f1 - f2),h)) . (k + 1)) . x = (((bdif (f1,h)) . (k + 1)) . x) - (((bdif (f2,h)) . (k + 1)) . x) ; :: thesis: S₁[k + 1]
let x be Real; :: thesis: ((bdif ((f1 - f2),h)) . ((k + 1) + 1)) . x = (((bdif (f1,h)) . ((k + 1) + 1)) . x) - (((bdif (f2,h)) . ((k + 1) + 1)) . x)
A6: ( ((bdif ((f1 - f2),h)) . (k + 1)) . x = (((bdif (f1,h)) . (k + 1)) . x) - (((bdif (f2,h)) . (k + 1)) . x) & ((bdif ((f1 - f2),h)) . (k + 1)) . (x - h) = (((bdif (f1,h)) . (k + 1)) . (x - h)) - (((bdif (f2,h)) . (k + 1)) . (x - h)) ) by A5;
A7: (bdif ((f1 - f2),h)) . (k + 1) is Function of REAL,REAL by Th12;
A8: (bdif (f2,h)) . (k + 1) is Function of REAL,REAL by Th12;
A9: (bdif (f1,h)) . (k + 1) is Function of REAL,REAL by Th12;
((bdif ((f1 - f2),h)) . ((k + 1) + 1)) . x = (bD (((bdif ((f1 - f2),h)) . (k + 1)),h)) . x by Def7
.= (((bdif ((f1 - f2),h)) . (k + 1)) . x) - (((bdif ((f1 - f2),h)) . (k + 1)) . (x - h)) by A7, Th4
.= ((((bdif (f1,h)) . (k + 1)) . x) - (((bdif (f1,h)) . (k + 1)) . (x - h))) - ((((bdif (f2,h)) . (k + 1)) . x) - (((bdif (f2,h)) . (k + 1)) . (x - h))) by A6
.= ((bD (((bdif (f1,h)) . (k + 1)),h)) . x) - ((((bdif (f2,h)) . (k + 1)) . x) - (((bdif (f2,h)) . (k + 1)) . (x - h))) by A9, Th4
.= ((bD (((bdif (f1,h)) . (k + 1)),h)) . x) - ((bD (((bdif (f2,h)) . (k + 1)),h)) . x) by A8, Th4
.= (((bdif (f1,h)) . ((k + 1) + 1)) . x) - ((bD (((bdif (f2,h)) . (k + 1)),h)) . x) by Def7
.= (((bdif (f1,h)) . ((k + 1) + 1)) . x) - (((bdif (f2,h)) . ((k + 1) + 1)) . x) by Def7 ;
hence ((bdif ((f1 - f2),h)) . ((k + 1) + 1)) . x = (((bdif (f1,h)) . ((k + 1) + 1)) . x) - (((bdif (f2,h)) . ((k + 1) + 1)) . x) ; :: thesis: verum

end;

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