
theorem thE1:
for R being Ring
for p being Polynomial of R
for i being Nat holds (<%0.R,1.R%> *' p).(i+1) = p.i
proof
let R be Ring, p be Polynomial of R; let i be Nat;
set q = <%0.R,1.R%>;
consider r being FinSequence of the carrier of R such that
A: len r = (i+1)+1 & (<%0.R,1.R%> *' p).(i+1) = Sum r &
   for k being Element of NAT st k in dom r
   holds r.k = q.(k-'1) * p.((i+1)+1-'k) by POLYNOM3:def 9;
B: dom r = Seg(i+2) by A,FINSEQ_1:def 3;
C: Seg 2 c= dom r by B,FINSEQ_1:5,NAT_1:11;
D: 2 in dom r by C,FINSEQ_1:3;
H2: 2 - 2 <= (i + 2) - 2 by NAT_1:11,XREAL_1:9;
H3: 0 <= 2 - 1;
E: r/.2 = r.2 by D,PARTFUN1:def 6
       .= q.(2-'1) * p.((i+1)+1-'2) by D,A
       .= q.1 * p.((i+1)+1-'2) by H3,XREAL_0:def 2
       .= q.1 * p.i by H2,XREAL_0:def 2
       .= 1.R * p.i by POLYNOM5:38;
F: now let k be Element of NAT;
   assume F: k in dom r & k <> 2; then
   per cases by XXREAL_0:1;
   suppose k < 2; then
    k + 1 <= 2 by INT_1:7; then
    F4: k + 1 - 1 <= 2 - 1 by XREAL_1:9;
    F5: 1 <= k by F,B,FINSEQ_1:1; then
    F3: k = 1 by F4,XXREAL_0:1;
    F2: k -' 1 = k - 1 by F5,XREAL_0:def 2;
    thus r/.k = r.k by F,PARTFUN1:def 6
             .= q.(k-'1) * p.((i+1)+1-'k) by A,F
             .= 0.R * p.((i+1)+1-'k) by F3,F2,POLYNOM5:38
             .= 0.R;
    end;
   suppose F4: k > 2; then
    2 + 1 <= k by INT_1:7; then
    F3: 2 + 1 - 1 <= k - 1 by XREAL_1:9;
    F2: k - 1 = k -' 1 by F4,XREAL_0:def 2;
    thus r/.k = r.k by F,PARTFUN1:def 6
             .= q.(k-'1) * p.((i+1)+1-'k) by A,F
             .= 0.R * p.((i+1)+1-'k) by F2,F3,POLYNOM5:38
             .= 0.R;
    end;
   end;
thus (<%0.R,1.R%> *' p).(i+1) = p.i by E,A,D,F,POLYNOM2:3;
end;
