
theorem Th8:
  for G being commutative Group,
  A,B being non empty finite set,
  FA be (the carrier of G)-valued total A -defined Function,
  FB be (the carrier of G)-valued total B -defined Function,
  FAB be (the carrier of G)-valued total A \/ B -defined Function
  st A misses B & FAB = FA +* FB holds
  Product (FAB) = (Product FA) * (Product FB)
  proof
    let G be commutative Group,
    A,B be non empty finite set,
    FA be (the carrier of G)-valued total A -defined Function,
    FB be (the carrier of G)-valued total B -defined Function,
    FAB be (the carrier of G)-valued total A \/ B -defined Function;
    assume A1: A misses B;
    assume A2: FAB = FA +* FB;
    consider fA being FinSequence of G such that
    A3:Product (FA) = Product fA & fA = FA*canFS(A) by Def1;
    consider fB being FinSequence of G such that
    A4:Product (FB) = Product fB & fB = FB*canFS(B) by Def1;
    set fAB = FAB*canFS(A \/ B);
    set cAB = canFS(A)^canFS(B);
    set uAB = canFS(A \/ B);
    reconsider SGAB = Seg (card(A \/ B)) as non empty finite set;
    A5: cAB is one-to-one onto FinSequence of (A \/ B) &
    dom (cAB) = SGAB by Lm3,A1;
    reconsider cAB as one-to-one onto FinSequence of (A \/ B) by Lm3,A1;
    rng (cAB) c= (A \/ B);
    then
    reconsider JcAB = (cAB) as Function of SGAB, (A \/ B) by FUNCT_2:2,A5;
    A6: dom uAB = Seg (len uAB) by FINSEQ_1:def 3
    .=SGAB by FINSEQ_1:93;
    rng (uAB) c= (A \/ B); then
    reconsider KuAB = uAB as Function of SGAB, (A \/ B) by FUNCT_2:2,A6;
    reconsider IuAB = (uAB)" as Function of (A \/ B), SGAB by FINSEQ_1:95;
    A7:rng uAB = (A \/ B) by FUNCT_2:def 3;
    then
    IuAB*KuAB = id SGAB & KuAB*IuAB = id (A \/ B) by FUNCT_2:29;
    then
    A8: IuAB is one-to-one & IuAB is onto by FUNCT_2:23;
    set p=IuAB*JcAB;
    p is onto & p is one-to-one by A8,FUNCT_2:27; then
    reconsider p as Permutation of SGAB;
    reconsider fAB as FinSequence of G by Lm2;
    A9: (canFS(A \/ B))*p = (KuAB * IuAB)*JcAB by RELAT_1:36
    .= id (A \/ B) * JcAB by A7,FUNCT_2:29
    .= canFS(A)^canFS(B) by FUNCT_2:17;
    A10: SGAB = dom fAB by Lm2;
    A11: fA ^ fB = FAB*(canFS(A) ^ canFS(B)) by A3,A4,Lm4,A1,A2
    .= fAB*p by RELAT_1:36,A9;
    thus Product (FAB) = Product (fAB) by Def1
    .= Product(fA ^ fB) by A10,A11,UPROOTS:16
    .= Product (FA)*Product (FB) by A3,A4,GROUP_4:5;
  end;
