defpred S₁[ Nat] means (id L) `^ $1 = id L;
IA: S₁[ 0 ] by T1;

IS: now :: thesis: for k being Nat st S₁[k] holds
S₁[k + 1]

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume IV: S₁[k] ; :: thesis: S₁[k + 1]
(id L) `^ (k + 1) = ((id L) `^ k) * (id L) by T3
.= id L by IV ;
hence S₁[k + 1] ; :: thesis: verum

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(IA, IS);
hence (id L) `^ n = id L ; :: thesis: verum