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

theorem Th1:
  for x be Function, g be Element of F.i holds
  ( dom x = I & x.i = g & for j be Element of I st j <> i holds x.j = 1_F.j )
  iff
  x = (1_product F)+*(i,g)
  proof
    let x be Function, g be Element of (F.i);
    A1: now assume
    A2: x = (1_product F)+* (i,g);
    A3: the carrier of product F = product Carrier F by GROUP_7:def 2;
    thus A4: dom x = dom (1_(product F)) by A2,FUNCT_7:30
    .= I by A3,PARTFUN1:def 2;
    dom x = dom (1_(product F)) by A2,FUNCT_7:30;
    hence x.i = g by A2,A4,FUNCT_7:31;
    thus for j be Element of I st j <> i holds x.j = 1_F.j
    proof
      let j be Element of I;
      assume j <> i; then
      x.j = (1_(product F)).j by A2,FUNCT_7:32;
      hence x.j = 1_F.j by GROUP_7:6;
    end;
  end;
  now assume A5:
    dom x = I & x.i = g & for j be Element of I st j <>i holds x.j = 1_F.j;
    the carrier of product F = product Carrier F by GROUP_7:def 2; then
    A6: dom (1_(product F)) = I by PARTFUN1:def 2;
    A7: dom ((1_(product F))+* (i,g))=dom x by A5,A6,FUNCT_7:30;
    set FG=(1_(product F))+* (i,g);
    now let j0 be object;
      assume j0 in dom x; then
      reconsider j=j0 as Element of I by A5;
      per cases;
      suppose A8: j <> i; then
        x.j = 1_F.j by A5;
        hence x.j0 = (1_(product F)).j by GROUP_7:6
        .= FG.j0 by A8,FUNCT_7:32;
      end;
      suppose j = i;
        hence x.j0 = FG.j0 by A6,A5,FUNCT_7:31;
      end;
    end;
    hence x = (1_(product F))+* (i,g) by A7,FUNCT_1:2;
  end;
  hence thesis by A1;
end;
