reserve x,y,z,X,Y for set;
reserve X,Y for non empty set,
  f for Function of X,Y;
reserve X, Y for non empty set,
  F for (BinOp of Y),
  B for (Element of Fin X),
  f for Function of X,Y;

theorem
  F is commutative & F is associative & F is idempotent implies for B,C
being Element of Fin X st B <> {} & B c= C holds F.(F$$(B,f),F$$(C,f)) = F$$(C,
  f)
proof
  assume
A1: F is commutative & F is associative & F is idempotent;
  let B,C be Element of Fin X such that
A2: B <> {} and
A3: B c= C;
  C <> {} by A2,A3;
  hence F.(F$$(B,f),F$$(C,f)) = F$$(B \/ C,f) by A1,A2,Th18
    .= F$$(C,f) by A3,XBOOLE_1:12;
end;
