reserve n for non zero Element of NAT;
reserve a,b,r,t for Real;

theorem Th34:
  for f be continuous PartFunc of REAL,REAL-NS n,
      F be PartFunc of REAL,REAL-NS n
       st a <= b
          & dom f =[' a,b '] & dom F =[' a,b ']
          & for t be Real st t in [.a,b.] holds F.t = integral(f,a,t)
   holds
 for x be Real st x in [.a,b.] holds F is_continuous_in x
proof
  let f be continuous PartFunc of REAL,REAL-NS n,
      F be PartFunc of REAL,REAL-NS n;
  assume A1: a <= b & dom f =[' a,b '] & dom F =[' a,b ']
          & for t be Real st t in [.a,b.]
                 holds F.t = integral(f,a,t);
  reconsider f0= f,F0= F as PartFunc of REAL, REAL n by REAL_NS1:def 4;
A2: f | ['a,b'] is bounded by A1,Th32;
  a in [.a,b.] by A1; then
A3: a in ['a,b'] by A1,INTEGRA5:def 3;
A4: f0  is continuous by NFCONT_4:23;
A5: now let t be Real;
    assume A6:t in [.a,b.];then
  A7:t in ['a,b'] by A1,INTEGRA5:def 3;
    thus F0.t = integral(f,a,t) by A1,A6
             .= integral(f0,a,t) by A1,A7,A2,Th33,A3,INTEGR19:48;
  end;
  now let x be Real;
    assume x in [.a,b.]; then
    F0 is_continuous_in x by A4,A5,A1,Th31;
    hence F is_continuous_in x by NFCONT_4:1;
  end;
  hence thesis;
end;
