Complete induction in the Peano system

Is complete induction valid in the Peano model of the naturals and why. In more detail if $L$ is the first order language $\{ +, \cdot, 0, <,S\}$ and $T$ is the theory with non-logical axioms

$$PA0.\qquad 0\not= SX_0$$ $$PA1.\qquad SX_0 = SX_1 \to X_0=X_1$$ $$PA2.\qquad X_0\cdot 0=0$$ $$PA3.\qquad X_0+SX_1=S(X_0+X_1)$$ $$PA4.\qquad X_0+0=X_0$$ $$PA5.\qquad X_0\cdot SX_1=X_0\cdot X_1+X_0$$ $$PA6.\qquad X_0 \not\lt X_0$$ $$PA7.\qquad X_0 \lt SX_1\equiv X_0=X_1 \lor X_0\lt X_1$$ $$PA8.\qquad X_0 \lt X_1 \lor X_0=X_1 \lor X_1 \lt X_0$$ $$PA9.\qquad X_0 \lt X_1 \to X_1 \lt X_0 \to X_0 \lt X_2$$

In addition, the induction scheme: if A is a formula from L than for each variable X $$PA10.\qquad A_X[0]\to\forall _X(A\to A_X[SX])\to\forall _XA$$

Than show that for all formulas A, for all variables X $$\vdash \forall _X(\forall _Y(Y\lt X\to A_X[Y])\to A)\to A$$

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