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 Th5:
  o is commutative implies (e is_a_unity_wrt o iff for a holds o.( a,e) = a)
proof
  assume
A1: o is commutative;
  now
    thus (for a holds o.(e,a) = a & o.(a,e) = a) implies for a holds o.(a,e) =
    a;
    assume
A2: for a holds o.(a,e) = a;
    let a;
    thus o.(e,a) = o.(a,e) by A1
      .= a by A2;
    thus o.(a,e) = a by A2;
  end;
  hence thesis by Th3;
end;
