reserve i,s,t,m,n,k for Nat,
        c,d,e for Element of NAT,
        fn for FinSequence of NAT,
        x,y for Integer;

theorem Th4:
  ex r be Nat st (1+s)|^t = 1 + t*s + (t choose 2)*s^2+r*s^3
proof
  defpred P[Nat] means
    ex r be Nat st (1+s)|^$1 = 1 + $1*s + ($1 choose 2)*s^2+r*s^3;
A1:   for t be Nat st P[t] holds P[t+1]
  proof let t be Nat;
      assume P[t];
      then consider r1 be Nat such that
A2:   (1+s)|^t = 1 + t*s + (t choose 2)*s^2 + r1*s^3;
      (1+s)|^(t+1)
    = (1 + t*s + (t choose 2)*s^2 + r1*s^3)*(1+s) by A2,NEWTON:6
   .= 1+(t+1)*s +(t+(t choose 2))*s^2+r1*s^3+(t choose 2)*s^2*s+r1*s^3*s
   .= 1+(t+1)*s +(t+t*(t-1)/2)*s^2+r1*s^3+(t choose 2)*s^2*s
           +r1*s^3*s by STIRL2_1:51
   .= 1+(t+1)*s +(t+1)*((t+1)-1)/2*s^2+r1*s^3+(t choose 2)*s^2*s+r1*s^3*s
   .=1+(t+1)*s +((t+1) choose 2)*s^2+r1*s^3+(t choose 2)*s^3+r1*s^3*s
                    by STIRL2_1:51
   .= 1+(t+1)*s +((t+1) choose 2)*s^2+(r1+(t choose 2)+r1*s)*s^3;
    hence thesis;
 end;
A3:P[0]
   proof
     reconsider z = 0 as Element of NAT;
     take r = 0;
     1 + z *s + (z choose 2) *s^2+r *s^3 = 1 + z *s + z *s^2+r *s^3
       by NEWTON:def 3
        .= (1+s)|^z by NEWTON:4;
     hence thesis;
   end;
  for t be Nat holds P[t] from NAT_1:sch 2(A3,A1);
  hence thesis;
end;
