reserve i,j,k,n,m,l,s,t for Nat;
reserve a,b for Real;
reserve F for real-valued FinSequence;
reserve z for Complex;
reserve x,y for Complex;
reserve r,s,t for natural Number;

theorem Th29:
  ((a,b) In_Power s).(s+1) = b|^s
proof
  reconsider m1 = s+1-1 as Element of NAT by INT_1:5,NAT_1:12;
  reconsider l1 = s-m1 as Element of NAT by INT_1:5;
  len ((a,b) In_Power s) = s+1 by Def4;
  then dom ((a,b) In_Power s) = Seg (s+1) by FINSEQ_1:def 3;
  then s+1 in dom ((a,b) In_Power s) by FINSEQ_1:4;
  then ((a,b) In_Power s).(s+1) = (s choose s)*a|^l1*b|^m1 by Def4
    .= 1*a|^l1*b|^m1 by Th21
    .= 1*b|^m1 by RVSUM_1:94
    .= b|^s;
  hence thesis;
end;
