reserve X for set;
reserve n,i for Element of NAT;
reserve a,b,c,d,e,r,x0 for Real;
reserve A for non empty closed_interval Subset of REAL;
reserve f,g,h for PartFunc of REAL,REAL n;
reserve E for Element of REAL n;

theorem
  for n be non zero Element of NAT,
      f be PartFunc of REAL,REAL-NS n
  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
  ex F be PartFunc of REAL,REAL-NS n 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 n be non zero Element of NAT;
    let f be PartFunc of REAL,REAL-NS n;
    consider F be PartFunc of REAL,REAL-NS n 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 Lm18;
    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,Th55;
    hence thesis by A1;
  end;
