
theorem
for A be non empty closed_interval Subset of REAL,
f be PartFunc of REAL,COMPLEX,
    g be Function of A,COMPLEX st f|A = g
  holds f is_integrable_on A iff g is integrable
proof
let A be non empty closed_interval Subset of REAL,
    f be PartFunc of REAL,COMPLEX,
    g be Function of A,COMPLEX;
assume A1: f|A = g;
thus f is_integrable_on A implies g is integrable
  proof
    assume A2: f is_integrable_on A;
    (Re g) is integrable & (Im g) is integrable
    proof
    A3:A = dom g by FUNCT_2:def 1
          .= dom f /\ A by A1,RELAT_1:61;
      then A = dom (Re f) /\ A by COMSEQ_3:def 3; then
      (Re f)||A is total by INTEGRA5:6,XBOOLE_1:17; then
      reconsider F = (Re f)|A as Function of A,REAL;
    A4:(Re f) is_integrable_on A by A2;
      dom g = A by FUNCT_2:def 1; then
      reconsider g0=g as PartFunc of REAL,COMPLEX by RELSET_1:5;
      Re g = Re g0
          .= F by A1,Lm4;
      hence (Re g) is integrable by A4;
      A = dom (Im f) /\ A by A3,COMSEQ_3:def 4; then
      (Im f)||A is total by INTEGRA5:6,XBOOLE_1:17; then
      reconsider G = (Im f)|A as Function of A,REAL;
    A5:(Im f) is_integrable_on A by A2;
      Im g = Im g0
          .= G by A1,Lm4;
      hence (Im g) is integrable by A5;
    end;
    hence thesis;
  end;
assume A6: g is integrable;
(Re f) is_integrable_on A & (Im f) is_integrable_on A
  by A1,Lm4,A6;
hence thesis;
end;
