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
  Sum^ A = 0 implies A = {} proof assume
A1: Sum^ A = 0 & A <> {}; then
    0 in dom A by ORDINAL3:8; then
    reconsider a = A.0 as Cantor-component Ordinal by Def11;
    0 in a & a c= Sum^ A by Th56,ORDINAL3:8;
    hence thesis by A1;
  end;
