Background: In the book Winning Ways for Your Mathematical Plays on combinatorial game theory, the Number Avoidance Theorem is stated:
DON'T MOVE IN A NUMBER UNLESS THERE'S NOTHING ELSE TO DO!
(sic.) The idea is that a move in a number guarantees a strictly worsened position for yourself. In the formal proof, they start:
It will be enough to prove that if $x$ is equal to a number, but $G$ is not, and if $$ G+x\mathop{\vert\triangleright}0, \quad \textrm{then some} \quad G^L+x\ge0. $$
My question: How should the theorem be precisely interpreted? My first interpretation was
(1) There is a move $G^L$ such that $G^L+x\ge G+x^L$.
Or might it be the (a priori) weaker statement
(2) If the moving player is winning in $G + x$, then there is a winning move from $G$.
(2) would mean that the player never misses out on a win by not playing in the number, but perhaps it might not be the "cleanest" win possible (i.e. with the highest values throughout). The reason for my confusion is that the given proof seems to prove (2), even though I thought (1) was the natural interpretation. So is (1) false, or are (1) and (2) in fact equivalent?