reserve
  a,b,c,d,e for Ordinal,
  m,n for Nat,
  f for Ordinal-Sequence,
  x for object;
reserve S,S1,S2 for Sequence;
reserve A,B for Cantor-normal-form Ordinal-Sequence;

theorem
  A <> {} implies ex c being Cantor-component Ordinal, B st A = <%c%>^B
  proof
    assume A <> {}; then
    consider n being Nat such that
A1: len A = n+1 by NAT_1:6;
    reconsider n as Element of NAT by ORDINAL1:def 12;
    n+1 = 1+^n by CARD_2:36; then
    consider S1,S2 such that
A2: A = S1^S2 & dom S1 = 1 & dom S2 = n by A1,Th2;
    reconsider S1,S2 as Ordinal-Sequence by A2,Th4;
    S1^S2 is Cantor-normal-form by A2; then
    reconsider S1,S2 as Cantor-normal-form Ordinal-Sequence by Th66;
    0 in Segm 1 by NAT_1:44; then
    reconsider c = S1.0 as Cantor-component Ordinal by A2,Def11;
    take c, S2;
    len S1 = 1 by A2;
    hence thesis by A2,AFINSQ_1:34;
  end;
