let f be PartFunc of REAL,REAL; :: thesis: for Z being Subset of REAL
for x being Real st x in Z holds
for n being Nat holds f . x = (Partial_Sums (Taylor (f,Z,x,x))) . n
let Z be Subset of REAL; :: thesis: for x being Real st x in Z holds
for n being Nat holds f . x = (Partial_Sums (Taylor (f,Z,x,x))) . n
let x be Real; :: thesis: ( x in Z implies for n being Nat holds f . x = (Partial_Sums (Taylor (f,Z,x,x))) . n )
assume A1: x in Z ; :: thesis: for n being Nat holds f . x = (Partial_Sums (Taylor (f,Z,x,x))) . n
defpred S₁[ Nat] means f . x = (Partial_Sums (Taylor (f,Z,x,x))) . $1;
A2: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A3: S₁[k] ; :: thesis: S₁[k + 1]
thus (Partial_Sums (Taylor (f,Z,x,x))) . (k + 1) = ((Partial_Sums (Taylor (f,Z,x,x))) . k) + ((Taylor (f,Z,x,x)) . (k + 1)) by SERIES_1:def 1
.= (f . x) + (((((diff (f,Z)) . (k + 1)) . x) * ((x - x) |^ (k + 1))) / ((k + 1) !)) by A3, Def7
.= (f . x) + (((((diff (f,Z)) . (k + 1)) . x) * ((0 |^ k) * 0)) / ((k + 1) !)) by NEWTON:6
.= f . x ; :: thesis: verum

end;

(Partial_Sums (Taylor (f,Z,x,x))) . 0 = (Taylor (f,Z,x,x)) . 0 by SERIES_1:def 1
.= ((((diff (f,Z)) . 0) . x) * ((x - x) |^ 0)) / (0 !) by Def7
.= (((f | Z) . x) * ((x - x) |^ 0)) / (0 !) by Def5
.= (((f | Z) . x) * 1) / 1 by NEWTON:4, NEWTON:12
.= f . x by A1, FUNCT_1:49 ;
then A4: S₁[ 0 ] ;
for k being Nat holds S₁[k] from NAT_1:sch 2(A4, A2);
hence for n being Nat holds f . x = (Partial_Sums (Taylor (f,Z,x,x))) . n ; :: thesis: verum