Cancellation Laws on Transfinite Cardinal Numbers

One of my papere discussed about transfinite cardinal number α, using Zorn’s lemma we have given a simple proof which is understandable to undergraduate students, of the result α + α = αα = α, that is, idempotency for addition and multiplication. Now moreover for a cardinal number b with 2 ≤ b < α we obtain easily α + b = αb = α, αb = αα = 2α, αb < bα. Using these results, we get many results directly as N0 + ∁ = N0∁ = ∁ + ∁ = ∁∁ = ∁, No No = ∁ = ∁No = 2No, ∁∁ = No∁ = 2∁ where N0 = card N, ∁ = card R.