多角形表記
だから、
( a ↑↑ b ) ↑↑ n ≒ a ↑ { ( a ↑↑ b ) ↑↑ ( n − 1 ) } {\displaystyle \left(a\uparrow \uparrow b\right)\uparrow \uparrow n\fallingdotseq a\uparrow \left\{\left(a\uparrow \uparrow b\right)\uparrow \uparrow \left(n-1\right)\right\}}
と近似してよい。
これを n が 1 になるまで繰り返せば、
( a ↑↑ b ) ↑↑ n ≒ a ↑ a ↑ ⋯ ↑ a ⏟ ( n − 1 ) copies of a ↑ { ( a ↑↑ b ) ↑↑ 1 } = a ↑ a ↑ ⋯ ↑ a ⏟ ( n − 1 ) copies of a ↑ ( a ↑↑ b ) ≒ a ↑↑ { ( n − 1 ) + b } {\displaystyle {\begin{aligned}\left(a\uparrow \uparrow b\right)\uparrow \uparrow n&\fallingdotseq \underbrace {a\uparrow a\uparrow \cdots \uparrow a} _{\left(n-1\right){\text{ copies of }}a}\uparrow \left\{\left(a\uparrow \uparrow b\right)\uparrow \uparrow 1\right\}\\&=\underbrace {a\uparrow a\uparrow \cdots \uparrow a} _{\left(n-1\right){\text{ copies of }}a}\uparrow \left(a\uparrow \uparrow b\right)\\&\fallingdotseq a\uparrow \uparrow \left\{\left(n-1\right)+b\right\}\end{aligned}}}