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 Th28:
  ((a,b) In_Power s).1 = a|^s
proof
  reconsider m1 = 1-1 as Element of NAT by INT_1:5;
  reconsider l1 = s-m1 as Nat;
  len ((a,b) In_Power s) = s+1 by Def4;
  then
A1: dom ((a,b) In_Power s) = Seg (s+1) by FINSEQ_1:def 3;
  s+1>=0+1 by XREAL_1:6;
  then 1 in dom ((a,b) In_Power s) by A1;
  then ((a,b) In_Power s).1 = (s choose 0)*a|^l1*b|^m1 by Def4
    .= 1*a|^l1*b|^m1 by Th19
    .= a|^s by RVSUM_1:94;
  hence thesis;
end;
