User blog:D57799/Another Theorem for Knuth Arrows

According to Knuth Arrows Theorem by Sbiis Saibian,

for all \(a≥2, b≥1, c≥1, k≥2: (a\uparrow^kb)\uparrow^kc(a\uparrow^k)^{b+c-3}(a+a\uparrow^ka)\)

\(=(a\uparrow^k)^{b+c-4}(a\uparrow^k(a+a\uparrow^ka))\)

\(>(a\uparrow^k)^{b+c-4}((a\uparrow^ka\uparrow^ka)\uparrow^ka)\)

\(>(a\uparrow^k)^{b+c-4}(a+a\uparrow^{k+1}2+a\uparrow^{k+1}3)\)

\(>......\)

\(>(a\uparrow^k)^{c-1}(a+a\uparrow^{k+1}2+a\uparrow^{k+1}3+......+a\uparrow^{k+1}b)\)

\(=(a\uparrow^k)^{c-2}(a\uparrow^k(a+a\uparrow^{k+1}2+a\uparrow^{k+1}3+......+a\uparrow^{k+1}b))\)

\(>(a\uparrow^k)^{c-2}((a\uparrow^{k-1})^{a\uparrow^{k+1}(b-1)+a\uparrow^{k+1}b}(a\uparrow^k(a+a\uparrow^{k+1}2+......+a\uparrow^{k+1}(b-2)))\)

\(>(a\uparrow^k)^{c-2}((a\uparrow^k(a\uparrow^{k+1}(b-1)+a\uparrow^{k+1}b)+a\uparrow^k(a+a\uparrow^{k+1}2+......+a\uparrow^{k+1}(b-2))\)

\(>(a\uparrow^k)^{c-2}((a\uparrow^k(a\uparrow^{k+1}(b-1)+a\uparrow^{k+1}b)+(((a\uparrow^k(a\uparrow^{k+1}(b-2)))\uparrow^k)...)\uparrow^ka\)

\(>(a\uparrow^k)^{c-2}((a\uparrow^{k+1})\uparrow^{k+1}2+a\uparrow^{k+1}(b-1)\times(b-1))\)

\(=(a\uparrow^k)^{c-2}(a+a\uparrow^{k+1}2+a\uparrow^{k+1}3+......+a\uparrow^{k+1}(b-1)+(a\uparrow^{k+1}b)))\uparrow^{k+1}2\)

\(>......\)

\(>a+a\uparrow^{k+1}2+a\uparrow^{k+1}3+......+a\uparrow^{k+1}(b-1)+(a\uparrow^{k+1}b)\uparrow^{k+1}c\)

\(>(a\uparrow^{k+1}b)\uparrow^{k+1}c\)