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;

theorem Th52:
  Sum^ {} = 0
  proof
   reconsider A = {} as finite Ordinal-Sequence;
   consider f being Ordinal-Sequence such that
A1:  Sum^ A = last f and
A2:  dom f = succ dom A and
A3:  f.0 = 0 and
  for n being Nat st n in dom A holds f.(n+1) = f.n +^ A.n by Def8;
     dom f = succ 0 implies last f = f.0 by ORDINAL2:6;
    hence Sum^ {} = 0 by A1,A2,A3;
  end;
