defpred S₁[ Nat] means ( P₁[$1] & P₁[$1 + 1] );
A4: S₁[ 0 ] by A1, A2;
A5: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A6: S₁[k] ; :: thesis: S₁[k + 1]
k + 2 = (k + 1) + 1 ;
hence S₁[k + 1] by A3, A6; :: thesis: verum

end;

A7: for k being Nat holds S₁[k] from NAT_1:sch 2(A4, A5);
let k be Nat; :: thesis: P₁[k]
thus P₁[k] by A7; :: thesis: verum