Theorem 7:
If {a,1,b}' is a limit ordinal, then r({a,1,b},n)' ≥ (a'+b')[n].
Proof:
If b = {0}, then
r({a,1,b},n)' = {{a,1,n},0,a}' = a'+n+1 ≥ a'+n = (a'+b')[n]
Otherwise, if assume theorem 7 holds for b.
r({a,1,b},n)' = {{a,1,r(b,n)},0,a}' = a'+r(b,n)'+1 ≥ a'+(b'[n]) = (a'+b')[n].
Theorem 8:
If a → 0, a' > 0, and {a,2,b}' is a limit ordinal, then r({a,2,b},n)' ≥ (a'*b')[n].
Proof:
If b = {0}, then
r({a,2,b},n)' = {{a,2,n},1,a}' = a'*n+a' ≥ a'*n = (a'*b')[n]
Otherwise, if assume theorem 8 holds for b.