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 Th66:
  M is commutative associative
implies
  M $$ ([#]dom <*f*>,A "**" <*f*>) = A "**" f
proof
  assume
A1: M is commutative associative;
A2: dom <*f*> = Seg 1 & 1 in {1} by FINSEQ_1:38,TARSKI:def 1;
  then reconsider O=1 as Element of dom <*f*> by FINSEQ_1:2;
  thus M $$ ([#]dom <*f*>, A "**" <*f*>) = (A "**" <*f*>).O
    by A1,SETWISEO:17,A2,FINSEQ_1:2
    .= <*A "**" f*>.O by Th62
    .= A "**" f;
end;
