
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
  integral(f,A) = integral(g)
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;
A2: 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;
dom g = A  by FUNCT_2:def 1; then
reconsider g0=g as PartFunc of REAL,COMPLEX by RELSET_1:5;
A3: Re g = Re g0
        .= F by A1,Lm4;
A = dom (Im f) /\ A by A2,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;
Im g = Im g0
    .= G by A1,Lm4;
hence integral(f,A) = integral(g) by A3;
end;
