let k be Nat; :: thesis: P₁[k]
defpred S₁[ Nat] means ( P₁[$1] & P₁[$1 + 1] );
A4: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
A5: k + 2 = (k + 1) + 1 ;
assume S₁[k] ; :: thesis: S₁[k + 1]
hence S₁[k + 1] by A3, A5; :: thesis: verum

end;

A6: S₁[ 0 ] by A1, A2;
for k being Nat holds S₁[k] from NAT_1:sch 2(A6, A4);
hence P₁[k] ; :: thesis: verum