
theorem lemf:
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) * (f`^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) * (f`^($1));
f`^(n+0) = (f`^n) * (id L) .= (f`^n) * (f`^0) by T1; then
IA: P[0];
IS: now let k be Nat;
    assume IV: P[k];
    f`^(n+(k+1)) = f`^((n+k)+1)
                .= ((f`^n) * (f`^k)) * f by IV,T3
                .= (f`^n) * ((f`^k) * f) by T2
                .= (f`^n) * (f`^(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;
