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;

theorem
  a in dom criticals f & b is_a_fixpoint_of f & (criticals f).a in b
  implies succ a in dom criticals f
  proof set g = criticals f;
    assume that
A1: a in dom g and
A2: b is_a_fixpoint_of f and
A3: g.a in b;
    consider c such that
A4: c in dom g & b = g.c by A2,Th33;
    a in c by A1,A3,A4,Th23; then
    succ a c= c by ORDINAL1:21;
    hence succ a in dom criticals f by A4,ORDINAL1:12;
  end;
