
theorem Th25:
  for I be non empty set,
      J be non-empty disjoint_valued ManySortedSet of I,
      F be Group-Family of I,J
  holds
    for x be Element of product(Union F), i be Element of I holds
    x | (J.i) = ((prod2dprod F).x).i
  proof
    let I be non empty set,
        J be non-empty disjoint_valued ManySortedSet of I,
        F be Group-Family of I,J;
    set f1 = dprod2prod F;
    set f2 = prod2dprod F;
    let x be Element of product(Union F), i be Element of I;
    A2: rng f1 = the carrier of product(Union F) by FUNCT_2:def 3;
    reconsider y = f2.x as Element of dprod F;
    ((prod2dprod F).x).i = f1.y | (J.i) by Def10
                        .= x | (J.i) by A2,FUNCT_1:35;
    hence thesis;
  end;
