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

theorem Th4:
  for g1 be Element of product F,
  z1 be Element of F.i st
  g1 = (1_product F)+* (i,z1) holds
  g1" = (1_product F)+* (i,z1")
  proof
    let g1 be Element of product F,
    z1 be Element of F.i;
    assume A1: g1=1_(product F)+* (i,z1);
    set x1=g1;
    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;
    set x12=g1";
    the carrier of product F = product Carrier F by GROUP_7:def 2; then
    A3: dom x12 = I by PARTFUN1:def 2;
    A4: x12.i = z1" by A2,GROUP_7:8;
    A5: for j be Element of I st i <> j holds x12.j = 1_F.j
    proof
      let j be Element of I;
      assume i <> j; then
      x1.j = 1_F.j by Th1,A1;
      hence x12.j = (1_F.j)" by GROUP_7:8
      .= 1_F.j by GROUP_1:8;
    end;
    thus thesis by A3,A4,A5,Th1;
  end;
