Cancellation laws: use only rules from 'basics' to prove cancellation laws for addition and multiplication for natural numbers. That is, prove for natural numbers m, n, k,

i) if m + k = n + k then m = n,

ii) if mk nk then m = n,

iii) if m + k

iv) if mk < nk then m

v) if m + k ≤n + k then m≤n,

vi) if mk ≤ nk then m≤ n.