code: def rs_mul(p1, p2, x, prec):
      """
      Return the product of the given two series, modulo ``O(x**prec)``.
  
      ``x`` is the series variable or its position in the generators.
  
      Examples
      ========
  
      >>> from sympy.polys.domains import QQ
      >>> from sympy.polys.rings import ring
      >>> from sympy.polys.ring_series import rs_mul
      >>> R, x = ring('x', QQ)
      >>> p1 = x**2 + 2*x + 1
      >>> p2 = x + 1
      >>> rs_mul(p1, p2, x, 3)
      3*x**2 + 3*x + 1
      """
      R = p1.ring
      p = R.zero
      if R.__class__ != p2.ring.__class__ or R != p2.ring:
          raise ValueError("p1 and p2 must have the same ring")
      iv = R.gens.index(x)
      if not isinstance(p2, PolyElement):
          raise ValueError("p2 must be a polynomial")
      if R == p2.ring:
          get = p.get
          items2 = list(p2.items())
          items2.sort(key=lambda e: e[0][iv])
          if R.ngens == 1:
              for exp1, v1 in p1.items():
                  for exp2, v2 in items2:
                      exp = exp1[0] + exp2[0]
                      if exp < prec:
                          exp = (exp,)
                          p[exp] = get(exp, 0) + v1 * v2
                      else:
                          break
          else:
              monomial_mul = R.monomial_mul
              for exp1, v1 in p1.items():
                  for exp2, v2 in items2:
                      if exp1[iv] + exp2[iv] < prec:
                          exp = monomial_mul(exp1, exp2)
                          p[exp] = get(exp, 0) + v1 * v2
                      else:
                          break
  
      p.strip_zero()
      return p
  
imports: from sympy.polys.domains import QQ, EX
  from sympy.polys.rings import PolyElement, ring, sring
  from sympy.polys.polyerrors import DomainError
  from sympy.polys.monomials import monomial_min, monomial_mul, monomial_div, monomial_ldiv
  from mpmath.libmp.libintmath import ifac
  from sympy.core import PoleError, Function, Expr
  from sympy.core.numbers import Rational
  from sympy.core.intfunc import igcd
  from sympy.functions import sin, cos, tan, atan, exp, atanh, tanh, log, ceiling
  from sympy.utilities.misc import as_int
  from mpmath.libmp.libintmath import giant_steps
  import math
label_desc: Use image node_id 7 for calling a global function with example usage: rs_mul(p1, p2, x, prec) and returns: p