
theorem
for c be Complex, A be non empty closed_interval Subset of REAL,
    f be PartFunc of REAL, COMPLEX
 st A c= dom f & f is_integrable_on A & f|A is bounded holds
  c(#)f is_integrable_on A & integral((c(#)f),A) = c * integral(f,A)
proof
let c be Complex;
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(c*integral(f,A)) = (Re c)*integral((Re f),A)-(Im c)*integral((Im f),A)
  & Im(c*integral(f,A)) = (Re c)*integral((Im f),A)+(Im c)*integral((Re f),A)
  proof
  A5: Re (integral(f,A)) = integral((Re f),A) by COMPLEX1:12;
  A6: Im (integral(f,A)) = integral((Im f),A) by COMPLEX1:12;
    hence Re(c*integral(f,A))
        = (Re c)*integral((Re f),A) - (Im c)*integral((Im f),A)
          by A5,COMPLEX1:9;
    thus Im(c*integral(f,A))
       = (Re c)*integral((Im f),A) + (Im c)*integral((Re f),A)
         by A5,A6,COMPLEX1:9;
  end;
(Re f)|A = Re (f|A) & (Im f)|A = Im (f|A) by Lm4; then
A7: (Re f)|A is bounded & (Im f)|A is bounded by A3,Th11;
A8: A c= dom (Re f) & A c= dom (Im f) by A1,COMSEQ_3:def 3,def 4;
A9: (Re f) is_integrable_on A & (Im f) is_integrable_on A by A2; then
A10: (Re c)(#)(Re f) is_integrable_on A & (Re c)(#)(Im f) is_integrable_on A
  & (Im c)(#)(Re f) is_integrable_on A
  & (Im c)(#)(Im f) is_integrable_on A by A7,A8,INTEGRA6:9;
A11: Re (c(#)f) = (Re c)(#)(Re f) - (Im c)(#)(Im f)
  & Im (c(#)f) = (Re c)(#)(Im f) + (Im c)(#)(Re f) by Th18;
A12: ((Re c)(#)(Re f))|A is bounded & ((Re c)(#)(Im f))|A is bounded
  & ((Im c)(#)(Re f))|A is bounded
  & ((Im c)(#)(Im f))|A is bounded by A7,RFUNCT_1:80;
dom (Re f) = dom f & dom (Im f) = dom f by COMSEQ_3:def 3,def 4; then
A13: A c= dom ((Re c)(#)(Re f)) & A c= dom ((Re c)(#)(Im f))
  & A c= dom ((Im c)(#)(Re f))
  & A c= dom ((Im c)(#)(Im f)) by A1,VALUED_1:def 5; then
A14: Re (c(#)f) is_integrable_on A by A10,A11,A12,INTEGRA6:11;
    Im (c(#)f) is_integrable_on A by A10,A11,A12,A13,INTEGRA6:11;
hence c(#)f is_integrable_on A by A14;
A15: Re (integral((c(#)f), A)) = integral(Re (c(#)f),A) by COMPLEX1:12
 .= integral((Re c)(#)(Re f),A) - integral((Im c)(#)(Im f),A)
    by A10,A11,A12,A13,INTEGRA6:11
 .= (Re c)*integral((Re f),A) - integral((Im c)(#)(Im f),A)
    by A9,A7,A8,INTEGRA6:9
 .= Re (c*integral(f,A)) by A4,A9,A7,A8,INTEGRA6:9;
Im (integral((c(#)f), A)) = integral( Im (c(#)f),A) by COMPLEX1:12
 .= integral((Re c)(#)(Im f),A)  + integral((Im c)(#)(Re f),A)
    by A10,A11,A12,A13,INTEGRA6:11
 .= (Re c)*integral((Im f),A) + integral((Im c)(#)(Re f),A)
    by A9,A7,A8,INTEGRA6:9
 .= Im (c*integral(f,A)) by A4,A9,A7,A8,INTEGRA6:9;
hence thesis by A15;
end;
