The tensor product expression is then abstracted by considering that if formula_35 and formula_44 represent "abstract basis vectors" from two sets formula_31 and formula_46, i.e. that "formula_47" and "formula_48", then pairs of these in the Cartesian product formula_49, i.e. formula_50 are taken as standing for the tensor products formula_23. (Note that the tensor products in the expression are in some sense "atomic", i.e. additions and scalar multiplications do not split them up into anything else, so we can replace them with something different without altering the mathematical structure.) With such an identification, we can thus define the tensor product of two free vector spaces formula_41 and formula_53 as being something (yet to be decided) that is isomorphic to formula_54.