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 Th32:
  for A, X, Y being non empty set for F being BinOp of A st F is
  idempotent & F is commutative & F is associative & F is having_a_unity for B
  being Element of Fin X for f being Function of X,Y holds for g being Function
  of Y,A holds F$$(f.:B,g) = F$$(B,g*f)
proof
  let A, X, Y be non empty set, F be BinOp of A such that
A1: F is idempotent and
A2: F is commutative & F is associative and
A3: F is having_a_unity;
  let B be Element of Fin X;
  let f be Function of X,Y;
  let g be Function of Y,A;
  now
    assume
A4: B = {};
    then f.:B = {}.Y;
    then
A5: F$$(f.:B,g) = the_unity_wrt F by A2,A3,Th28;
    B = {}.X by A4;
    hence thesis by A2,A3,A5,Th28;
  end;
  hence thesis by A1,A2,Th26;
end;
