reserve a,b,c,d for Ordinal;
reserve l for non empty limit_ordinal Ordinal;
reserve u for Element of l;
reserve A for non empty Ordinal;
reserve e for Element of A;
reserve X,Y,x,y,z for set;
reserve n,m for Nat;
reserve f for Ordinal-Sequence;
reserve U,W for Universe;
reserve F,phi for normal Ordinal-Sequence of W;
reserve g for Ordinal-Sequence-valued Sequence;

theorem Th53:
  for g st dom g <> {} & for a st a in dom g holds g.a is normal
  holds criticals g is continuous
  proof
    let g;
    assume A1: dom g <> {};
    assume A2: for a st a in dom g holds g.a is normal;
    set f = criticals g;
    let a,b;
    reconsider h = f|a as increasing Ordinal-Sequence by ORDINAL4:15;
    assume
A3: a in dom f & a <> 0 & a is limit_ordinal & b = f.a; then
A4: b = Union h by A1,A2,Th52;
    a c= dom f by A3,ORDINAL1:def 2; then
    dom h = a by RELAT_1:62;
    hence b is_limes_of f|a by A3,A4,ORDINAL5:6;
  end;
