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 Th56:
  for A being finite Ordinal-Sequence holds A.0 c= Sum^ A
  proof
    let A be finite Ordinal-Sequence;
    defpred P[finite Sequence] means
    for A being finite Ordinal-Sequence st $1 = A holds A.0 c= Sum^ A;
A1: P[{}];
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;
      per cases;
      suppose f = {}; then
        g = {}^<%x%> by A5;
        then g = <%x%>;
        then g.0 = x & Sum^ g = x by Th53;
        hence g.0 c= Sum^ g;
      end;
      suppose f <> {}; then
        0 in dom f9 & Sum^ g = Sum^ f9 +^ x by A5,Th54,ORDINAL3:8; then
        g.0 = f9.0 & f9.0 c= Sum^ f9 & Sum^ f9 c= Sum^ g
        by A3,A5,ORDINAL3:24,ORDINAL4:def 1;
        hence g.0 c= Sum^ g;
      end;
    end;
    for f being finite Sequence holds P[f] from AFINSQ_1:sch 3(A1,A2);
    hence A.0 c= Sum^ A;
  end;
