reserve x,y,y1,y2 for set,
  D for non empty set,
  d,d1,d2,d3 for Element of D,
  F,G,H,H1,H2 for FinSequence of D,
  f,f1,f2 for sequence of D,
  g for BinOp of D,
  k,n,i,l for Nat,
  P for Permutation of dom F;

theorem
  g is associative commutative & (g is having_a_unity or len F >= 1) &
  len F = len G & len F = len H &
  (for k being Nat st k in dom F holds F.k = g.(G.k,H.k))
  implies g "**" F = g.(g "**" G,g "**" H)
proof
A1: dom F = Seg len F & dom G = Seg len G by FINSEQ_1:def 3;
A2: len F = 0 or len F >= 1 by NAT_1:14;
  assume g is associative commutative &( g is having_a_unity or len F >= 1);
  hence thesis by A1,A2,Lm12,Lm13;
end;
