
theorem Th9:
  for G being non empty multMagma,
  q be set,
  z be Element of G,
  f be (the carrier of G)-valued total {q}-defined Function
  st f = q .--> z
  holds Product f = z
  proof
    let G be non empty multMagma,
    q be set,
    z be Element of G,
    f be (the carrier of G)-valued total {q}-defined Function;
    assume A1: f = q .--> z;
    set zz = <*z*>;
    rng zz = {z} by FINSEQ_1:38; then
    reconsider zz as FinSequence of G by FINSEQ_1:def 4;
    A2: f* canFS({q}) is FinSequence of G
    & dom (f*canFS({q})) = Seg card({q}) by Lm2;
    reconsider g= f* canFS({q}) as FinSequence of G by Lm2;
    A3: card {q} = 1 by CARD_1:30;
    then dom (g) = Seg 1 by Lm2;
    then
    A4: len g = 1 by FINSEQ_1:def 3;
    A5:canFS({q}) = <*q*> by FINSEQ_1:94;
    A6: q in {q} by TARSKI:def 1;
    A7: 1 in dom g by A2,A3;
    g.1 = f.((canFS({q})).1) by FUNCT_1:12,A7
    .=f.q by A5
    .=z by A1,FUNCOP_1:7,A6; then
    <*z*> = f* canFS({q}) by A4,FINSEQ_1:40;
    hence Product f = Product zz by Def1
    .= z by GROUP_4:9;
  end;
