reserve X for BCI-algebra;
reserve x,y,z for Element of X;
reserve i,j,k,l,m,n for Nat;
reserve f,g for sequence of the carrier of X;

theorem Th10:
  Polynom (m,n+1,x,y) = Polynom (m,n,x,y) \ (y\x)
proof
  consider f such that
A1: ((x,(x\y)) to_power (m+1),(y\x)) to_power (n+1) = f.(n+1) and
A2: f.0 = (x,(x\y)) to_power (m+1) and
A3: for k st k < n+1 holds f.(k + 1) = f.k \ (y\x) by BCIALG_2:def 1;
  consider g such that
A4: ((x,(x\y)) to_power (m+1),(y\x)) to_power n = g.n and
A5: g.0 = (x,(x\y)) to_power (m+1) and
A6: for k st k < n holds g.(k + 1) = g.k \ (y\x) by BCIALG_2:def 1;
  defpred P[Nat] means $1 <= n implies f.$1=g.$1;
  now
    let k;
    assume
A7: k<=n implies f.k=g.k;
    set m=k+1;
    assume
A8: m<=n;
    then k +1 <= n+1 by NAT_1:13;
    then k<n+1 by NAT_1:13;
    then
A9: f.(k + 1) = f.k \(y\x) by A3;
    k<n by A8,NAT_1:13;
    hence f.m=g.m by A6,A7,A9;
  end;
  then
A10: for k st P[k] holds P[k+1];
A11: P[0] by A2,A5;
  for n holds P[n] from NAT_1:sch 2(A11,A10);
  then f.n = g.n;
  hence thesis by A1,A3,A4,XREAL_1:29;
end;
