code: def test_R3():
      a, b, c = symbols("a b c", positive=True)
      m = Matrix([[a], [b], [c]])
  
      assert m == R3_c.transform(
          R3_r, R3_r.transform(R3_c, m)
      ).applyfunc(simplify)
      # TODO assert m == R3_r.transform(R3_c, R3_c.transform(R3_r, m)).applyfunc(simplify)
      assert m == R3_s.transform(
          R3_r, R3_r.transform(R3_s, m)
      ).applyfunc(simplify)
      # TODO assert m == R3_r.transform(R3_s, R3_s.transform(R3_r, m)).applyfunc(simplify)
      assert m == R3_s.transform(
          R3_c, R3_c.transform(R3_s, m)
      ).applyfunc(simplify)
      # TODO assert m == R3_c.transform(R3_s, R3_s.transform(R3_c, m)).applyfunc(simplify)
  
      with warns_deprecated_sympy():
          assert m == R3_c.coord_tuple_transform_to(
              R3_r, R3_r.coord_tuple_transform_to(R3_c, m)
          ).applyfunc(simplify)
          # TODO assert m == R3_r.coord_tuple_transform_to(R3_c, R3_c.coord_tuple_transform_to(R3_r, m)).applyfunc(simplify)
          assert m == R3_s.coord_tuple_transform_to(
              R3_r, R3_r.coord_tuple_transform_to(R3_s, m)
          ).applyfunc(simplify)
          # TODO assert m == R3_r.coord_tuple_transform_to(R3_s, R3_s.coord_tuple_transform_to(R3_r, m)).applyfunc(simplify)
          assert m == R3_s.coord_tuple_transform_to(
              R3_c, R3_c.coord_tuple_transform_to(R3_s, m)
          ).applyfunc(simplify)
  
imports: from sympy.core import Lambda, Symbol, symbols
  from sympy.diffgeom.rn import R2, R2_p, R2_r, R3_r, R3_c, R3_s, R2_origin
  from sympy.diffgeom import Manifold, Patch, CoordSystem, Commutator, Differential, TensorProduct, WedgeProduct, BaseCovarDerivativeOp, CovarDerivativeOp, LieDerivative, covariant_order, contravariant_order, twoform_to_matrix, metric_to_Christoffel_1st, metric_to_Christoffel_2nd, metric_to_Riemann_components, metric_to_Ricci_components, intcurve_diffequ, intcurve_series
  from sympy.simplify import trigsimp, simplify
  from sympy.functions import sqrt, atan2, sin
  from sympy.matrices import Matrix
  from sympy.testing.pytest import raises, nocache_fail
  from sympy.testing.pytest import warns_deprecated_sympy
label_desc: Use image node_id 3 for calling a global function with example usage: test_R3() without return types