Suppose I = R. Then obviously, as 1 R, also 1 I.

Clearly, 1 is an invertible element of I.

Assume that v is an element of I with inverse r. Then 1 = rv is an expression as required in Assertion 4.

Suppose that Assertion 4 holds: there are v₁, ..., v_n I and r₁, ..., r_n R such that 1 = r₁, ..., r_n I such that 1 = r₁v₁+ ··· + r_nv_n. By the theorem on the previous page, the right-hand side belongs to I. As this expression is equal to 1, the latter also belongs to I.