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 Th25:
  1 in a & dom f c= omega & (for b st b in dom f holds f.b = a|^|^b)
  implies f is increasing
  proof assume that
A1: 1 in a and
A2: dom f c= omega and
A3: for n being Ordinal st n in dom f holds f.n = a|^|^n;
    let b,c; assume
A4: b in c & c in dom f; then
A5: b in dom f by ORDINAL1:10;
    reconsider b,c as Element of omega by A2,A4,ORDINAL1:10;
    b in Segm c by A4;
    then f.b = a|^|^b & f.c = a|^|^c & b < c by A3,A4,A5,NAT_1:44;
    hence thesis by A1,Th24;
  end;
