
theorem Th45:
  for I be non empty set,
      G be Group,
      x be finite-support Function of I,G,
      a be Element of G
  st I = {1,2} & x = <*a,1_G*>
  holds Product x = a
  proof
    let I be non empty set,
        G be Group,
        x be finite-support Function of I,G,
        a be Element of G;
    assume
    A1: I = {1,2} & x = <*a,1_G*>;
    reconsider i1 = 1 as Element of I by A1,TARSKI:def 2;
    set y = (I --> 1_G) +* (i1,a);
    A2: dom y = dom (I --> 1_G) by FUNCT_7:30
             .= I by FUNCOP_1:13;
    A3: dom (I --> 1_G) = I by FUNCOP_1:13;
    for i be object  st i in dom x holds x.i = y.i
    proof
      let i be object;
      assume
      A4: i in dom x; then
      A5: i = 1 or i = 2 by A1,TARSKI:def 2;
      reconsider i0 = i as Element of I by A4;
      per cases;
      suppose
        A6: i = 1;
        thus x.i = a by A1,A6
                .= y.i by A3,A6,FUNCT_7:31;
      end;
      suppose
        A7: i <> 1;
        hence y.i = (I --> 1_G).i by FUNCT_7:32
                .= 1_G by A4,FUNCOP_1:7
                .= x.i by A1,A5,A7;
      end;
    end; then
    x = (I --> 1_G) +* (i1,a) by A2,FUNCT_2:def 1;
    hence thesis by GROUP_19:21;
  end;
