reserve a,b,c,d,e,x,r for Real,
  A for non empty closed_interval Subset of REAL,
  f,g for PartFunc of REAL,REAL;

theorem
  for x0 be 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 be PartFunc of REAL,REAL st ].a,b.[ c= dom F & (for x be Real
st x in ].a,b.[ holds F.x = integral(f,a,x)) & F is_differentiable_in x0 & diff
  (F,x0)=f.x0
proof
  let x0 be Real;
  consider F be PartFunc of REAL,REAL such that
A1: ].a,b.[ c= dom F & for x be 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;
  then F is_differentiable_in x0 & diff(F,x0)=f.x0 by A1,Th28;
  hence thesis by A1;
end;
