Exact estimates for provability of the rule of transfinite induction in initial parts of arithmetic

It is possible to transform any deduction of a prenex formula in $Z_n$ into a deduction of the same formula in $Z_n$ consisting of (sequents consisting of) prenex formulas. By obvious further transformations it is possible to put all quantifier formulas into prenex form with matrix being an equality and prefix consisting of changing qunatifiers ($Qx_1\bar{Q}x_2Qx_3\bar{Q}x_4\dots$).The grade (number of logical connectives) of such a formula is greater by unity than its complexity. Applying some modification of Gentzen reduction one is able to prove by transfinite induction (TI) that it is possible to transform $Z_n$-deduction of arbitrary closed formula $\exists x(r(x)=0)$ in its $Z_0$-deduction. It is possible to assigne ordinals to deductions in such a way that the ordinal of $Z_n$-proof is less than $\omega_{n+3}(\omega=0,\omega_{i+1}=\omega^{\omega_i})$. So every function provably recursive in $Z_n$ is definable by transfinite recursion on some ordinal $<\omega_{n+3}$, and there is a quantifier-free example of the rule TI up to $\omega_{n+3}$ underivable in $Z_n$. On the other hand inspection of Gentzen's 1943 proof of derivability of TI (this proof needs a refinement: quantifiers are to be taken into consideration) shows that the rule TI up to $\omega_{n+2}$ is derivable in $Z_n$. It is noted that the complexity of a deduction could be measured by the complexity of induction formulas: if all induction formulas in a deduction are of the complexity $\leq n$, then all cut formulas could be made of the complexity $\leq n$.