defpred S₁[ Nat] means f `^ R is isomorphism ;
f `^ 0 = id R by T1;
then IA: S₁[ 0 ] ;

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]
f `^ (k + 1) = (f `^ k) * f by T3;
hence S₁[k + 1] by IV; :: thesis: verum

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(IA, IS);
hence for b₁ being Function of R,R st b₁ = f `^ n holds
b₁ is RingIsomorphism ; :: thesis: verum