reserve x,y for object,
        D,D1,D2 for non empty set,
        i,j,k,m,n for Nat,
        f,g for FinSequence of D*,
        f1 for FinSequence of D1*,
        f2 for FinSequence of D2*;
reserve f for complex-valued Function,
        g,h for complex-valued FinSequence;

theorem Th23:
  for r,s be Real holds
    r |^ <*s*> = <* r to_power s *>
proof
  let r,s be Real;
  A1: len <*s*> =1 by FINSEQ_1:39;
  dom <*s*> = Seg 1 & Seg 1 = {1} by FINSEQ_1:2,38;
  then 1 in dom <*s*> & <*s*>.1=s;
  then (r |^ <*s*>).1 = r to_power s by Def4;
  hence thesis by A1,CARD_1:def 7,FINSEQ_1:40;
end;
