reserve I for non empty set,
  J for ManySortedSet of I,
  S for non void non empty ManySortedSign,
  i for Element of I,
  c for set,
  A for MSAlgebra-Family of I,S,
  EqR for Equivalence_Relation of I,
  U0,U1,U2 for MSAlgebra over S,
  s for SortSymbol of S,
  o for OperSymbol of S,
  f for Function;

theorem Th1:
  for f,F being Function, A being set st f in product F holds f|A
  in product(F|A)
proof
  let f,F be Function, A be set;
  assume
A1: f in product F;
  then dom f = dom F by CARD_3:9;
  then
A2: dom (f|A) = dom F /\ A by RELAT_1:61
    .= dom (F|A)by RELAT_1:61;
  for x be object st x in dom (F|A) holds (f|A).x in (F|A).x
  proof
    let x be object;
    assume
A3: x in dom (F|A);
    then x in dom F /\ A by RELAT_1:61;
    then
A4: x in dom F by XBOOLE_0:def 4;
    (F|A).x = F.x & (f|A).x = f.x by A2,A3,FUNCT_1:47;
    hence thesis by A1,A4,CARD_3:9;
  end;
  hence thesis by A2,CARD_3:9;
end;
