
theorem Th5:
  for I be non empty set,
      F be multMagma-Family of I,
      x be Function,
      i be Element of I
  st x in product F
  holds x.i in F.i
  proof
    let I be non empty set;
    let F be multMagma-Family of I;
    let x be Function;
    let i be Element of I;
    assume
    A1: x in product F;
    A2: (Carrier F).i = [#](F.i) by PENCIL_3:7;
    i in I; then
    A4: i in dom(Carrier F) by PARTFUN1:def 2;
    x in product Carrier F by A1,GROUP_7:def 2;
    hence thesis by A2,A4,CARD_3:9;
  end;
