User blog comment:Triakula/FGH and SGH at large ordinals/@comment-35470197-20191214094756/@comment-35470197-20191214115637

> FGH and SGH is like defined here and here.

I meant "Which one (f or g) stands for FGH?" but anyway please never mind it, because I used FGH_α and SGH_α.

> I can't argue, but can it be proven generally?

If I am correct, the following is a proof:
 * Let α be an ordinal equipped with a system of fundamental sequences for ordinals below α+1. I show that FGH_β(x) > SGH_β(x) for any β≦α and x > 1 by transfinite induction. By inductioon hypothesis, FGH_γ(x) > SGH_γ(x) for any γ<β and x > 1. If β = 0, then we have FGH_β(x) = x+1 > 0 = SGH_β(x) for any x > 1. If β is a successor ordinal δ+1, then FGH_β(x) = FGH_δ^x(x) > FGH_δ^2(x) > FGH_δ(x)+1 > SGH_δ(x)+1 = SGH_β(x) for any x > 1 by the induction hypothesis applied to δ < β. If β is a non-zero limit ordinal, we have FGH_β(x) = FGH_{β[x]}(x) > SGH_{β[x]}(x) = SGH_β(x) for any x > 1 by the induction hypothesis applied to β[x] < β. Thus we obtain FGH_β(x) > SGH_β(x) for any x > 1. In particular, we obtain FGH_α(x) > SGH_α(x) for any x > 1.