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 Th13:
  for A being non empty set, o,o9 being BinOp of A holds o9 is
commutative implies (o9 is_distributive_wrt o iff for a,b,c being Element of A
  holds o9.(o.(a,b),c) = o.(o9.(a,c),o9.(b,c)))
proof
  let A be non empty set, o,o9 be BinOp of A;
  assume
A1: o9 is commutative;
  (for a,b,c being Element of A holds o9.(a,o.(b,c)) = o.(o9.(a,b),o9.(a,c
  )) & o9.(o.(a,b),c) = o.(o9.(a,c),o9.(b,c))) iff for a,b,c being Element of A
  holds o9.(o.(a,b),c) = o.(o9.(a,c),o9.(b,c))
  proof
    thus (for a,b,c being Element of A holds o9.(a,o.(b,c)) = o.(o9.(a,b),o9.(
a,c)) & o9.(o.(a,b),c) = o.(o9.(a,c),o9.(b,c))) implies for a,b,c being Element
    of A holds o9.(o.(a,b),c) = o.(o9.(a,c),o9.(b,c));
    assume
A2: for a,b,c being Element of A holds o9.(o.(a,b),c) = o.(o9.(a,c),o9 .(b,c));
    let a,b,c be Element of A;
    thus o9.(a,o.(b,c)) = o9.(o.(b,c),a) by A1
      .= o.(o9.(b,a),o9.(c,a)) by A2
      .= o.(o9.(a,b),o9.(c,a)) by A1
      .= o.(o9.(a,b),o9.(a,c)) by A1;
    thus thesis by A2;
  end;
  hence thesis by Th11;
end;
