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;
reserve x,A for set;

theorem Th37:
  (|.f.|)|A = |. f|A .|
proof
  dom((|.f.|)|A) = dom |.f.| /\ A by RELAT_1:61;
  then
A1: dom((|.f.|)|A) = dom f /\ A by VALUED_1:def 11;
A2: dom(f|A) = dom f /\ A by RELAT_1:61;
  then
A3: dom|. f|A .| = dom f /\ A by VALUED_1:def 11;
  now
    let x be Element of X;
    assume
A4: x in dom((|.f.|)|A);
    then (|.(f|A).|).x = |. (f|A).x .| by A1,A3,VALUED_1:def 11;
    then
A5: (|.(f|A).|).x = |. f.x .| by A2,A1,A4,FUNCT_1:47;
    x in dom f by A1,A4,XBOOLE_0:def 4;
    then
A6: x in dom |.f.| by VALUED_1:def 11;
    ((|.f.|)|A).x = (|.f.|).x by A4,FUNCT_1:47;
    hence ((|.f.|)|A).x = (|. f|A .|).x by A6,A5,VALUED_1:def 11;
  end;
  hence thesis by A1,A3,PARTFUN1:5;
end;
