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 Th18:
  g is associative & (g is having_a_unity or k <> 0 & l <> 0)
  implies g "**" ((k + l) |-> d) = g.(g "**" (k |-> d),g "**" (l |-> d))
proof
A1: k <> 0 & l <> 0 implies len(k |-> d) >= 1 & len(l |-> d) >= 1 by NAT_1:14;
  (k + l) |-> d = (k |-> d) ^ (l |-> d) by FINSEQ_2:123;
  hence thesis by A1,Th5;
end;
