let n be Element of NAT ; :: thesis: for f being PartFunc of REAL ,REAL
for r being Real st 0 < r & ].(- r),r.[ c= dom f & f is_differentiable_on n + 1,].(- r),r.[ holds
for x being Real st x in ].(- r),r.[ holds
ex s being Real st
( 0 < s & s < 1 & abs ((f . x) - ((Partial_Sums (Maclaurin f,].(- r),r.[,x)) . n)) = abs (((((diff f,].(- r),r.[) . (n + 1)) . (s * x)) * (x |^ (n + 1))) / ((n + 1) ! )) )
let f be PartFunc of REAL ,REAL ; :: thesis: for r being Real st 0 < r & ].(- r),r.[ c= dom f & f is_differentiable_on n + 1,].(- r),r.[ holds
for x being Real st x in ].(- r),r.[ holds
ex s being Real st
( 0 < s & s < 1 & abs ((f . x) - ((Partial_Sums (Maclaurin f,].(- r),r.[,x)) . n)) = abs (((((diff f,].(- r),r.[) . (n + 1)) . (s * x)) * (x |^ (n + 1))) / ((n + 1) ! )) )
let r be Real; :: thesis: ( 0 < r & ].(- r),r.[ c= dom f & f is_differentiable_on n + 1,].(- r),r.[ implies for x being Real st x in ].(- r),r.[ holds
ex s being Real st
( 0 < s & s < 1 & abs ((f . x) - ((Partial_Sums (Maclaurin f,].(- r),r.[,x)) . n)) = abs (((((diff f,].(- r),r.[) . (n + 1)) . (s * x)) * (x |^ (n + 1))) / ((n + 1) ! )) ) )
assume A1: ( 0 < r & ].(- r),r.[ c= dom f & f is_differentiable_on n + 1,].(- r),r.[ ) ; :: thesis: for x being Real st x in ].(- r),r.[ holds
ex s being Real st
( 0 < s & s < 1 & abs ((f . x) - ((Partial_Sums (Maclaurin f,].(- r),r.[,x)) . n)) = abs (((((diff f,].(- r),r.[) . (n + 1)) . (s * x)) * (x |^ (n + 1))) / ((n + 1) ! )) )
let x be Real; :: thesis: ( x in ].(- r),r.[ implies ex s being Real st
( 0 < s & s < 1 & abs ((f . x) - ((Partial_Sums (Maclaurin f,].(- r),r.[,x)) . n)) = abs (((((diff f,].(- r),r.[) . (n + 1)) . (s * x)) * (x |^ (n + 1))) / ((n + 1) ! )) ) )
assume x in ].(- r),r.[ ; :: thesis: ex s being Real st
( 0 < s & s < 1 & abs ((f . x) - ((Partial_Sums (Maclaurin f,].(- r),r.[,x)) . n)) = abs (((((diff f,].(- r),r.[) . (n + 1)) . (s * x)) * (x |^ (n + 1))) / ((n + 1) ! )) )
then ex s being Real st
( 0 < s & s < 1 & f . x = ((Partial_Sums (Maclaurin f,].(- r),r.[,x)) . n) + (((((diff f,].(- r),r.[) . (n + 1)) . (s * x)) * (x |^ (n + 1))) / ((n + 1) ! )) ) by A1, Th2;
hence ex s being Real st
( 0 < s & s < 1 & abs ((f . x) - ((Partial_Sums (Maclaurin f,].(- r),r.[,x)) . n)) = abs (((((diff f,].(- r),r.[) . (n + 1)) . (s * x)) * (x |^ (n + 1))) / ((n + 1) ! )) ) ; :: thesis: verum