let n be Nat; :: thesis: for f being PartFunc of REAL,REAL
for Z being Subset of REAL
for b, l being Real ex g being Function of REAL,REAL st
for x being Real holds g . x = ((f . b) - ((Partial_Sums (Taylor (f,Z,x,b))) . n)) - ((l * ((b - x) |^ (n + 1))) / ((n + 1) !))

let f be PartFunc of REAL,REAL; :: thesis: for Z being Subset of REAL
for b, l being Real ex g being Function of REAL,REAL st
for x being Real holds g . x = ((f . b) - ((Partial_Sums (Taylor (f,Z,x,b))) . n)) - ((l * ((b - x) |^ (n + 1))) / ((n + 1) !))

let Z be Subset of REAL; :: thesis: for b, l being Real ex g being Function of REAL,REAL st
for x being Real holds g . x = ((f . b) - ((Partial_Sums (Taylor (f,Z,x,b))) . n)) - ((l * ((b - x) |^ (n + 1))) / ((n + 1) !))

let b, l be Real; :: thesis: ex g being Function of REAL,REAL st
for x being Real holds g . x = ((f . b) - ((Partial_Sums (Taylor (f,Z,x,b))) . n)) - ((l * ((b - x) |^ (n + 1))) / ((n + 1) !))

deffunc H1( Real) -> Element of REAL = In ((((f . b) - ((Partial_Sums (Taylor (f,Z,\$1,b))) . n)) - ((l * ((b - \$1) |^ (n + 1))) / ((n + 1) !))),REAL);
consider g being Function of REAL,REAL such that
A1: for d being Element of REAL holds g . d = H1(d) from take g ; :: thesis: for x being Real holds g . x = ((f . b) - ((Partial_Sums (Taylor (f,Z,x,b))) . n)) - ((l * ((b - x) |^ (n + 1))) / ((n + 1) !))
let x be Real; :: thesis: g . x = ((f . b) - ((Partial_Sums (Taylor (f,Z,x,b))) . n)) - ((l * ((b - x) |^ (n + 1))) / ((n + 1) !))
reconsider x = x as Element of REAL by XREAL_0:def 1;
g . x = H1(x) by A1;
hence g . x = ((f . b) - ((Partial_Sums (Taylor (f,Z,x,b))) . n)) - ((l * ((b - x) |^ (n + 1))) / ((n + 1) !)) ; :: thesis: verum