defpred S₁[ Nat] means f |^ $1 is linear-transformation of V1,V1;
f |^ 0 = id V1 by Th18;
then A1: S₁[ 0 ] ;
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 S₁[k] ; :: thesis: S₁[k + 1]
then reconsider fk = f |^ k as linear-transformation of V1,V1 ;
f |^ (k + 1) = fk * (f |^ 1) by Th20
.= fk * f by Th19 ;
hence S₁[k + 1] ; :: thesis: verum

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(A1, A2);
hence f |^ n is linear-transformation of V1,V1 ; :: thesis: verum