reserve A for set;
reserve X,Y,Z for set,x,x1,x2,y,y1,y2,z,z1,z2 for object;
reserve u for UnOp of A,
  o,o9 for BinOp of A,
  a,b,c,e,e1,e2 for Element of A;

theorem
  for A being non empty set, o,o9 being BinOp of A holds o9 is
  commutative implies (o9 is_right_distributive_wrt o iff o9
  is_left_distributive_wrt o)
proof
  let A be non empty set, o,o9 be BinOp of A;
  assume
A1: o9 is commutative;
  then o9 is_distributive_wrt o iff o9 is_left_distributive_wrt o by Th14;
  hence thesis by A1,Th15;
end;
