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 Th45:
  for phi being Ordinal-Sequence st
     for c st c in dom phi holds phi.c = epsilon_c
  holds phi is increasing
  proof let f; assume
A1: for c st c in dom f holds f.c = epsilon_c;
    let a,b; assume
A2: a in b & b in dom f; then
    f.a = epsilon_a & f.b = epsilon_b by A1,ORDINAL1:10;
    hence thesis by A2,Th44;
  end;
