
theorem t4a:
for R being domRing
for a being non zero Element of R
for n being Nat holds <%0.R,a%>`^n = (a|^n) * (<%0.R,1.R%>`^n)
proof
let R be domRing, a be non zero Element of R, n be Nat;
defpred P[Nat] means <%0.R,a%>`^($1) = (a|^($1)) * (<%0.R,1.R%>`^($1));
    <%0.R,a%>`^0 = (1_R) * (1_.(R)) by POLYNOM5:15
                .= (a|^0) * (1_.(R)) by BINOM:8
                .= (a|^0) * (<%0.R,1.R%>`^0) by POLYNOM5:15; then
IA: P[0];
IS: now let k be Nat;
    assume IV: P[k];
    <%0.R,a%>`^(k+1)
        = (<%0.R,a%>`^k) *' <%0.R,a%> by POLYNOM5:19
       .= ((a|^k) * (<%0.R,1.R%>`^k)) *' (a * <%0.R,1.R%>) by IV,t4b
       .= a * (((a|^k) * (<%0.R,1.R%>`^k)) *' <%0.R,1.R%>) by RING_4:10
       .= a * ((a|^k) * ((<%0.R,1.R%>`^k) *' <%0.R,1.R%>)) by RING_4:10
       .= (a * (a|^k)) * ((<%0.R,1.R%>`^k) *' <%0.R,1.R%>) by RING_4:11
       .= ((a|^k) * a|^1) * ((<%0.R,1.R%>`^k) *' <%0.R,1.R%>) by BINOM:8
       .= (a|^(k+1)) * ((<%0.R,1.R%>`^k) *' <%0.R,1.R%>) by BINOM:10;
    hence P[k+1] by POLYNOM5:19;
    end;
for k being Nat holds P[k] from NAT_1:sch 2(IA,IS);
hence thesis;
end;
