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
  1 in a & (for b st b in dom f holds f.b = exp(a,b)) implies f is increasing
  proof assume
A1: 1 in a & for b st b in dom f holds f.b = exp(a,b);
    let b,d; assume
A2: b in d & d in dom f; then
    f.b = exp(a,b) & f.d = exp(a,d) by A1,ORDINAL1:10;
    hence thesis by A1,A2,ORDINAL4:24;
  end;
