reserve
  I for non empty set,
  F for associative Group-like multMagma-Family of I,
  i, j for Element of I;

theorem Th3:
  for g1,g2 be Element of product F,
  z1,z2 be Element of F.i st
  g1 = (1_(product F))+* (i,z1) &
  g2 = (1_(product F))+* (i,z2) holds
  g1 * g2 = (1_product F)+* (i,z1*z2)
  proof
    let g1,g2 be Element of product F,
    z1,z2 be Element of F.i;
    assume A1: g1=((1_(product F))+* (i,z1)) &
    g2=((1_(product F))+* (i,z2));
    set x1=g1, x2 = g2;
    A2: x1=g1 & dom x1 = I & x1.i = z1 &
    for j be Element of I st j <> i holds x1.j = 1_F.j by Th1,A1;
    A3: x2=g2 & dom x2 = I & x2.i = z2 &
    for j be Element of I st j <> i holds x2.j = 1_F.j by Th1,A1;
    set x12=g1*g2;
    the carrier of product F = product Carrier F by GROUP_7:def 2; then
    A4: dom x12 = I by PARTFUN1:def 2;
    A5: x12.i = z1*z2 by A2,A3,GROUP_7:1;
    for j be Element of I st i <> j holds x12.j = 1_F.j
    proof
      let j be Element of I;
      assume A6: i <> j; then
A7:   x1.j = 1_F.j by Th1,A1;
      x2.j = 1_F.j by A6,Th1,A1;
      hence x12.j = 1_F.j*1_F.j by A7,GROUP_7:1
      .= 1_F.j by GROUP_1:def 4;
    end;
    hence thesis by A4,A5,Th1;
  end;
