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 Th27:
  (a,b) In_Power 0 = <*1*>
proof
  set F = (a,b) In_Power 0;
A1: len F = 0+1 by Def4
    .= 1;
  then
A2: dom F = {1} by FINSEQ_1:2,def 3;
  then 1 in dom F by TARSKI:def 1;
  then consider i be set such that
A3: i in dom F;
  reconsider i as Element of NAT by A3;
A4: i = 1 by A2,A3,TARSKI:def 1;
  then reconsider m1 = i-1 as Element of NAT by INT_1:5;
  reconsider l1 = 0-m1 as Element of NAT by A4;
  1 in dom (a,b) In_Power 0 by A2,TARSKI:def 1;
  then F.1 = (0 choose 0)*a|^l1*b|^m1 by A4,Def4
    .= 1*a|^0*b|^0 by A4,Th21
    .= 1 by RVSUM_1:94;
  hence thesis by A1,FINSEQ_1:40;
end;
