reserve X for RealNormSpace;

theorem Th14:
  for A be non empty closed_interval Subset of REAL,
      f1, f2 be PartFunc of REAL,the carrier of X
        st f1 is_integrable_on A & f2 is_integrable_on A &
        A c= dom f1 & A c= dom f2 holds
        f1 + f2 is_integrable_on A &
        integral(f1 + f2,A) = integral(f1,A) + integral(f2,A)
proof
  let A be non empty closed_interval Subset of REAL;
  let f1, f2 be PartFunc of REAL,the carrier of X;
  assume that
A1: f1 is_integrable_on A &
    f2 is_integrable_on A and
A2: A c= dom f1 & A c= dom f2;
  A c= dom f1 /\ dom f2 by A2,XBOOLE_1:19; then
A3: A c= dom(f1 + f2) by VFUNCT_1:def 1;
  consider g1 be Function of A,the carrier of X such that
A4: g1 = f1|A & g1 is integrable by A1;
  consider g2 be Function of A,the carrier of X such that
A5: g2 = f2|A & g2 is integrable by A1;
  (f1 + f2)|A = f1|A + f2|A by VFUNCT_1:27; then
A6: (f1 + f2)|A = g1 + g2 by A4,A5,Th10;
  g1 + g2 is total by VFUNCT_1:32; then
  reconsider g = g1 + g2 as Function of A,the carrier of X;
  g is integrable by Th6,A4,A5;
  hence (f1 + f2) is_integrable_on A by A6;
  thus integral(f1 + f2,A) = integral(g) by Def8,A6,A3
                          .= integral(g1) + integral(g2) by Th6,A4,A5
                          .= integral(f1,A) + integral(g2) by A2,A4,Def8
                          .= integral(f1,A) + integral(f2,A) by A2,A5,Def8;
end;
