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 Th55:
  for A being finite Ordinal-Sequence holds Sum^ (<%a%>^A) = a +^ Sum^ A
  proof
    defpred P[finite Sequence] means
    for f being finite Ordinal-Sequence st f = $1
    holds Sum^(<%a%>^f) = a +^ Sum^ f;
    Sum^(<%a%>^{}) = a by Th53; then
A1: P[{}] by Th52,ORDINAL2:27;
A2: for f being finite Sequence, x being object st P[f] holds P[f^<%x%>]
    proof
      let f be finite Sequence;
      let x be object; assume
A3:   P[f];
      let g be finite Ordinal-Sequence;
      consider b such that
A4:   rng g c= b by ORDINAL2:def 4;
      assume
A5:   g = f^<%x%>; then
      rng g = (rng f)\/rng<%x%> by AFINSQ_1:26; then
A6:   rng f c= b & rng <%x%> c= b by A4,XBOOLE_1:11; then
      reconsider f9 = f as finite Ordinal-Sequence by ORDINAL2:def 4;
      rng <%x%> = {x} by AFINSQ_1:33; then
      x in b by A6,ZFMISC_1:31; then
      reconsider x as Ordinal;
      thus Sum^(<%a%>^g) = Sum^ (<%a%>^f9^<%x%>) by A5,AFINSQ_1:27
      .= Sum^ (<%a%>^f9) +^ x by Th54
      .= a +^ Sum^ f9 +^ x by A3
      .= a +^ (Sum^ f9 +^ x) by ORDINAL3:30
      .= a +^ Sum^ g by A5,Th54;
    end;
    for f being finite Sequence holds P[f] from AFINSQ_1:sch 3(A1,A2);
    hence thesis;
  end;
