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
  0 in a & (for b st b in dom f holds f.b = a|^|^b) implies f is non-decreasing
  proof assume that
A1: 0 in a and
A2: for b st b in dom f holds f.b = a|^|^b;
    let b,c; assume
A3: b in c & c in dom f; then
    b c= c by ORDINAL1:def 2; then
    a|^|^b c= a|^|^c & a|^|^c = f.c by A1,A2,A3,Th21;
    hence f.b c= f.c by A2,A3,ORDINAL1:10;
  end;
