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;
