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 & f . x = ((Partial_Sums (Maclaurin f,].(- r),r.[,x)) . n) + (((((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 & f . x = ((Partial_Sums (Maclaurin f,].(- r),r.[,x)) . n) + (((((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 & f . x = ((Partial_Sums (Maclaurin f,].(- r),r.[,x)) . n) + (((((diff f,].(- r),r.[) . (n + 1)) . (s * x)) * (x |^ (n + 1))) / ((n + 1) ! )) ) )
assume that
A1: 0 < r and
A0: ].(- r),r.[ c= dom f and
A2: 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 & f . x = ((Partial_Sums (Maclaurin f,].(- r),r.[,x)) . n) + (((((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 & f . x = ((Partial_Sums (Maclaurin f,].(- r),r.[,x)) . n) + (((((diff f,].(- r),r.[) . (n + 1)) . (s * x)) * (x |^ (n + 1))) / ((n + 1) ! )) ) )
assume A3: x in ].(- r),r.[ ; :: thesis: 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) ! )) )
consider s being Real such that
A4: ( 0 < s & s < 1 ) and
A5: f . x = ((Partial_Sums (Taylor f,].(0 - r),(0 + r).[,0 ,x)) . n) + (((((diff f,].(0 - r),(0 + r).[) . (n + 1)) . (0 + (s * (x - 0 )))) * ((x - 0 ) |^ (n + 1))) / ((n + 1) ! )) by A1, A2, A3, A0, TAYLOR_1:33;
take s ; :: thesis: ( 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) ! )) )
thus ( 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 A4, A5; :: thesis: verum