
theorem
for r be Real for A be non empty closed_interval Subset of REAL
for f be PartFunc of REAL, COMPLEX
  st A c= dom f & f is_integrable_on A
  & f|A is bounded holds
    r(#)f is_integrable_on A
  & integral((r(#)f),A) = r * integral(f,A)
proof
let r be Real;
let A be non empty closed_interval Subset of REAL;
let f be PartFunc of REAL, COMPLEX;
assume that A1: A c= dom f and
A2: f is_integrable_on A and
A3: f|A is bounded;
A4: (Re f) is_integrable_on A & (Im f) is_integrable_on A by A2;
A5: integral(r(#)(Re f), A)=r*integral((Re f), A)
  & integral(r(#)(Im f), A)=r*integral((Im f), A)
  proof
    Re (f|A) is bounded & Im (f|A) is bounded by A3,Th11; then
  A6: (Re f)|A is bounded & (Im f)|A is bounded by Lm4;
  A7: A c= dom (Re f) & A c= dom (Im f) by A1,COMSEQ_3:def 3,def 4;
    hence integral(r(#)(Re f),A)=r*integral((Re f),A) by A4,A6,INTEGRA6:9;
    thus integral(r(#)(Im f),A)=r*integral((Im f),A) by A4,A6,A7,INTEGRA6:9;
  end;
A8: Re (integral((r(#)f), A)) = r * Re (integral(f,A))
   & Im (integral((r(#)f), A)) = r * Im (integral(f,A))
  proof
    Re (integral((r(#)f),A)) = integral((Re (r(#)f)),A)
  & r*Re (integral(f,A)) = r*integral((Re f),A)
  & Im (integral((r(#)f),A)) = integral((Im (r(#)f)),A)
  & r*Im (integral(f,A)) = r*integral((Im f),A) by COMPLEX1:12;
    hence thesis by A5,MESFUN6C:2;
  end;
(Re (r(#)f)) is_integrable_on A & (Im (r(#)f)) is_integrable_on A
  proof
    (Re f)|A = Re (f|A) & (Im f)|A = Im (f|A) by Lm4; then
  A9:(Re f)|A is bounded & (Im f)|A is bounded by A3,Th11;
  A10:A c= dom (Re f) & A c= dom (Im f) by A1,COMSEQ_3:def 3,def 4;
    (Re f) is_integrable_on A & (Im f) is_integrable_on A by A2; then
    r(#)(Re f) is_integrable_on A & r(#)(Im f ) is_integrable_on A
    by A9,A10,INTEGRA6:9;
    hence thesis by MESFUN6C:2;
  end;
hence r(#)f is_integrable_on A;
  Re (integral((r(#)f), A)) = Re (r*integral(f,A))
& Im (integral((r(#)f), A)) = Im (r*integral(f,A)) by A8,Th1;
hence thesis;
end;
