
theorem lemID:
for R being Ring
for S being R-homomorphic Ring
for f being multiplicative unity-preserving Function of R,S
for a being Element of R
for n being Nat holds f.(a|^n) = (f.a)|^n
proof
let R be Ring, S be R-homomorphic Ring;
let f be multiplicative unity-preserving Function of R,S;
let a be Element of R, n be Nat;
defpred P[Nat] means f.(a|^($1)) = (f.a)|^($1);
f.(a|^0) = f.(1_R) by BINOM:8
        .= 1_S by GROUP_1:def 13 .= (f.a)|^0 by BINOM:8; then
IA: P[0];
IS: now let k be Nat;
    assume IV: P[k];
    f.(a|^(k+1))
         = f.((a|^k) * a|^1) by BINOM:10
        .= f.(a|^k) * f.(a|^1) by GROUP_6:def 6
        .= (f.a)|^k * f.a by IV,BINOM:8
        .= (f.a)|^k * (f.a)|^1 by BINOM:8
        .= (f.a)|^(k+1) by BINOM:10;
    hence P[k+1];
    end;
for k being Nat holds P[k] from NAT_1:sch 2(IA,IS);
hence thesis;
end;
