let n be Nat; :: thesis: for a being non trivial Nat holds Py (a,n) >= n
let a be non trivial Nat; :: thesis: Py (a,n) >= n
defpred S₁[ Nat] means Py (a,$1) >= $1;
A1: S₁[ 0 ] ;
A2: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
k + 1 > k by NAT_1:13;
then A3: (Py (a,k)) + 1 <= Py (a,(k + 1)) by Th14, NAT_1:13;
assume S₁[k] ; :: thesis: S₁[k + 1]
then k + 1 <= (Py (a,k)) + 1 by XREAL_1:6;
hence S₁[k + 1] by A3, XXREAL_0:2; :: thesis: verum

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(A1, A2);
hence Py (a,n) >= n ; :: thesis: verum