reserve i,j,n,k,m for Nat,
     a,b,x,y,z for object,
     F,G for FinSequence-yielding FinSequence,
     f,g,p,q for FinSequence,
     X,Y for set,
     D for non empty set;
reserve
  B,A,M for BinOp of D,
  F,G for D* -valued FinSequence,
  f for FinSequence of D,
  d,d1,d2 for Element of D;
reserve
  F,G for non-empty non empty FinSequence of D*,
  f for non empty FinSequence of D;

theorem Th65:
  M is commutative associative
implies
  M $$ ([#]dom (F^G), A "**" (F^G)) =
    M.(M $$ ([#]dom F, A "**" F), M $$ ([#]dom G, A "**" G))
proof
A1: dom (A "**" (F^G) ) = dom (F^G)
    & dom (A "**" F)=dom F & dom (A "**" G)=dom G by FUNCT_2:def 1;
  A "**" (F^G) = (A "**" F) ^ (A "**" G) by Th63;
  hence thesis by A1,Th64;
end;
