let a, b be Real; :: thesis: for f being PartFunc of REAL,REAL

for x0 being Real st a <= b & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & x0 in ].a,b.[ & f is_continuous_in x0 holds

ex F being PartFunc of REAL,REAL st

( ].a,b.[ c= dom F & ( for x being Real st x in ].a,b.[ holds

F . x = integral (f,a,x) ) & F is_differentiable_in x0 & diff (F,x0) = f . x0 )

let f be PartFunc of REAL,REAL; :: thesis: for x0 being Real st a <= b & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & x0 in ].a,b.[ & f is_continuous_in x0 holds

ex F being PartFunc of REAL,REAL st

( ].a,b.[ c= dom F & ( for x being Real st x in ].a,b.[ holds

F . x = integral (f,a,x) ) & F is_differentiable_in x0 & diff (F,x0) = f . x0 )

let x0 be Real; :: thesis: ( a <= b & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & x0 in ].a,b.[ & f is_continuous_in x0 implies ex F being PartFunc of REAL,REAL st

( ].a,b.[ c= dom F & ( for x being Real st x in ].a,b.[ holds

F . x = integral (f,a,x) ) & F is_differentiable_in x0 & diff (F,x0) = f . x0 ) )

consider F being PartFunc of REAL,REAL such that

A1: ( ].a,b.[ c= dom F & ( for x being Real st x in ].a,b.[ holds

F . x = integral (f,a,x) ) ) by Lm13;

assume ( a <= b & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & x0 in ].a,b.[ & f is_continuous_in x0 ) ; :: thesis: ex F being PartFunc of REAL,REAL st

( ].a,b.[ c= dom F & ( for x being Real st x in ].a,b.[ holds

F . x = integral (f,a,x) ) & F is_differentiable_in x0 & diff (F,x0) = f . x0 )

then ( F is_differentiable_in x0 & diff (F,x0) = f . x0 ) by A1, Th28;

hence ex F being PartFunc of REAL,REAL st

( ].a,b.[ c= dom F & ( for x being Real st x in ].a,b.[ holds

F . x = integral (f,a,x) ) & F is_differentiable_in x0 & diff (F,x0) = f . x0 ) by A1; :: thesis: verum

for x0 being Real st a <= b & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & x0 in ].a,b.[ & f is_continuous_in x0 holds

ex F being PartFunc of REAL,REAL st

( ].a,b.[ c= dom F & ( for x being Real st x in ].a,b.[ holds

F . x = integral (f,a,x) ) & F is_differentiable_in x0 & diff (F,x0) = f . x0 )

let f be PartFunc of REAL,REAL; :: thesis: for x0 being Real st a <= b & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & x0 in ].a,b.[ & f is_continuous_in x0 holds

ex F being PartFunc of REAL,REAL st

( ].a,b.[ c= dom F & ( for x being Real st x in ].a,b.[ holds

F . x = integral (f,a,x) ) & F is_differentiable_in x0 & diff (F,x0) = f . x0 )

let x0 be Real; :: thesis: ( a <= b & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & x0 in ].a,b.[ & f is_continuous_in x0 implies ex F being PartFunc of REAL,REAL st

( ].a,b.[ c= dom F & ( for x being Real st x in ].a,b.[ holds

F . x = integral (f,a,x) ) & F is_differentiable_in x0 & diff (F,x0) = f . x0 ) )

consider F being PartFunc of REAL,REAL such that

A1: ( ].a,b.[ c= dom F & ( for x being Real st x in ].a,b.[ holds

F . x = integral (f,a,x) ) ) by Lm13;

assume ( a <= b & f is_integrable_on ['a,b'] & f | ['a,b'] is bounded & ['a,b'] c= dom f & x0 in ].a,b.[ & f is_continuous_in x0 ) ; :: thesis: ex F being PartFunc of REAL,REAL st

( ].a,b.[ c= dom F & ( for x being Real st x in ].a,b.[ holds

F . x = integral (f,a,x) ) & F is_differentiable_in x0 & diff (F,x0) = f . x0 )

then ( F is_differentiable_in x0 & diff (F,x0) = f . x0 ) by A1, Th28;

hence ex F being PartFunc of REAL,REAL st

( ].a,b.[ c= dom F & ( for x being Real st x in ].a,b.[ holds

F . x = integral (f,a,x) ) & F is_differentiable_in x0 & diff (F,x0) = f . x0 ) by A1; :: thesis: verum