reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  f,g for PartFunc of X,COMPLEX,
  r for Real,
  k for Real,
  n for Nat,
  E for Element of S;

theorem Th30:
  for f,g be PartFunc of X,REAL holds (R_EAL f)(#)(R_EAL g) = R_EAL(f(#)g)
proof
  let f,g be PartFunc of X,REAL;
A1: dom ((R_EAL f)(#)(R_EAL g)) = dom(R_EAL f) /\ dom(R_EAL g)
    by MESFUNC1:def 5;
A2: now
    let x be Element of X;
    assume
A3: x in dom((R_EAL f)(#)(R_EAL g));
    then x in dom(f(#)g) by A1,VALUED_1:def 4;
    then
A4: (f(#)g).x = f.x * g.x by VALUED_1:def 4
    .= (f.x) qua ExtReal * g.x;
    ((R_EAL f)(#)(R_EAL g)).x = (R_EAL f).x * (R_EAL g).x by A3,MESFUNC1:def 5;
    hence ((R_EAL f)(#)(R_EAL g)).x = (R_EAL(f(#)g)).x by A4;
  end;
  dom ((R_EAL f)(#)(R_EAL g)) = dom R_EAL(f(#)g) by A1,VALUED_1:def 4;
  hence thesis by A2,PARTFUN1:5;
end;
