theorem
  ( ex A be Element of S st A = dom f & f is A-measurable ) & dom f =
dom g & g is_integrable_on M & ( for x be Element of X st x in dom f holds
  |.f.x qua Complex.| <= g.x ) implies
  f is_integrable_on M & Integral(M,abs f) <= Integral(M,g)
proof
  assume that
A1: ex A be Element of S st A = dom f & f is A-measurable and
A2: dom f = dom g and
A3: g is_integrable_on M and
A4: for x be Element of X st x in dom f holds |.f.x qua Complex.| <= g.x;
A5: R_EAL g is_integrable_on M by A3;
A6: now
    let x be Element of X;
A7: |.f.x qua Complex.| = |. (R_EAL f).x .| by EXTREAL1:12;
    assume x in dom R_EAL f;
    hence |. (R_EAL f).x .| <= (R_EAL g).x by A4,A7;
  end;
  consider A be Element of S such that
A8: A = dom f and
A9: f is A-measurable by A1;
  R_EAL f is A-measurable by A9;
  then R_EAL f is_integrable_on M & Integral(M,|.(R_EAL f).|) <= Integral(M,
  R_EAL g ) by A2,A8,A5,A6,MESFUNC5:102;
  hence thesis by Th1;
end;
