
theorem
for L being non empty 1-sorted
for f being Function of L,L
for n,m being Nat holds f`^(n*m) = (f`^n)`^m
proof
let L be non empty 1-sorted, f be Function of L,L, n,m be Nat;
defpred P[Nat] means f`^(n*($1)) = (f`^n)`^($1);
f`^(n*0) = id L by T1 .= (f`^n)`^0 by T1; then
IA: P[0];
IS: now let k be Nat;
    assume IV: P[k];
    f`^(n*(k+1)) = f`^(n*k+n)
                .= ((f`^n)`^k) * (f`^n) by IV,lemf
                .= (f`^n)`^(k+1) by T3;
    hence P[k+1];
    end;
for k being Nat holds P[k] from NAT_1:sch 2(IA,IS);
hence thesis;
end;
