
theorem Th48:
  for G be Group, a be FinSequence of G
  st for i be object st i in dom a holds a.i = 1_G
  holds Product a = 1_G
  proof
    let G be Group, a be FinSequence of G;
    assume
    A1: for i be object st i in dom a holds a.i = 1_G;
    set n = len a;
    a = n |-> 1_G
    proof
      n |-> 1_G = Seg n --> 1_G by FINSEQ_2:def 2; then
      A2: dom(n |-> 1_G) = Seg n by FUNCOP_1:13;
      A3: dom a = Seg n by FINSEQ_1:def 3;
      for i be object st i in dom a holds a.i = (n |-> 1_G).i
      proof
        let i be object;
        assume
        A4: i in dom a; then
        i in Seg n by FINSEQ_1:def 3; then
        (n |-> 1_G).i = 1_G by FINSEQ_2:57;
        hence thesis by A1,A4;
      end;
      hence thesis by A2,A3,FUNCT_1:2;
    end; then
    Product a = (1_G) |^ n by GROUP_4:12
             .= 1_G by GROUP_1:31;
    hence thesis;
  end;
