
theorem Th49:
  for I be finite set,
      G be Group,
      a be (the carrier of G)-valued total I-defined Function
  st for i be object st i in I holds a.i = 1_G
  holds Product a = 1_G
  proof
    let I be finite set,
        G be Group,
        a be (the carrier of G)-valued total I-defined Function;
    assume
    A1: for i be object st i in I holds a.i = 1_G;
    set cs = canFS(I);
    set acs = a * cs;
    A2: rng acs c= the carrier of G;
    A3: I = dom a & rng cs = I by FUNCT_2:def 3,PARTFUN1:def 2; then
    dom acs = dom cs by RELAT_1:27; then
    dom acs = Seg len cs by FINSEQ_1:def 3; then
    acs is FinSequence by FINSEQ_1:def 2; then
    reconsider acs as FinSequence of G by A2,FINSEQ_1:def 4;
    A4: Product a = Product acs by GROUP_17:def 1;
    for i be object st i in dom acs holds acs.i = 1_G
    proof
      let i be object;
      assume
      A5: i in dom acs; then
      i in dom cs by A3,RELAT_1:27; then
      cs.i in rng cs by FUNCT_1:3; then
      a.(cs.i) =1_G by A1;
      hence thesis by A5,FUNCT_1:12;
    end;
    hence thesis by A4,Th48;
  end;
