let n be Nat; :: thesis: Ackermann (3,n) = (2 |^ (n + 3)) - 3
defpred S₁[ Nat] means H₁(3,$1) = (2 |^ ($1 + 3)) - 3;
A1: S₁[ 0 ]

proof

thus H₁(3, 0 ) = H₁(3 - 1,1) by Def1
.= (2 * 1) + 3 by Th10
.= (2 |^ (0 + 3)) - 3 by Lm2 ; :: thesis: verum

end;

A2: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A3: S₁[k] ; :: thesis: S₁[k + 1]
thus H₁(3,k + 1) = H₁(3 - 1,H₁(3,(k + 1) - 1)) by Def1
.= (2 * H₁(3,k)) + 3 by Th10
.= (2 * (2 |^ (k + 3))) - 3 by A3
.= (2 |^ ((k + 3) + 1)) - 3 by NEWTON:6
.= (2 |^ ((k + 1) + 3)) - 3 ; :: thesis: verum

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(A1, A2);
hence Ackermann (3,n) = (2 |^ (n + 3)) - 3 ; :: thesis: verum