theorem Th37:
  f is_integrable_on M implies Integral(M,f) in COMPLEX &
  Integral(M,|.f.|) in REAL & |.f.| is_integrable_on M
proof
  assume
A1: f is_integrable_on M;
  reconsider AF = |.f.| as PartFunc of X,REAL;
  AF is_integrable_on M by A1,MESFUN7C:35;
  hence thesis by LPSPACE1:44,XCMPLX_0:def 2;
end;
