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 Th53:
  Sum^ <%a%> = a
  proof
    consider f being Ordinal-Sequence such that
A1: Sum^ <%a%> = last f & dom f = succ dom <%a%> & f.0 = 0 &
    for n being Nat st n in dom <%a%> holds f.(n+1) = f.n +^ <%a%>.n by Def8;
A2: dom <%a%> = 1 & <%a%>.0 = a by AFINSQ_1:def 4;
    0 in Segm 1 by NAT_1:44; then
    f.(0+1) = (0 qua Ordinal) +^ a by A1,A2 .= a by ORDINAL2:30;
    hence thesis by A1,A2,ORDINAL2:6;
  end;
