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
  succ a in dom criticals f & b is_a_fixpoint_of f & (criticals f).a in b
  implies (criticals f).succ a c= b
  proof set g = criticals f;
    set Y = {c where c is Element of dom f: c is_a_fixpoint_of f};
    set X = ord-type Y;
    assume
A1: succ a in dom g & b is_a_fixpoint_of f & g.a in b; then
    consider c such that
A2: c in dom g & b = g.c by Th33;
    a in succ a by ORDINAL1:6; then
A3: a in dom g & g.a in g.succ a by A1,ORDINAL1:10,ORDINAL2:def 12;
    On Y = Y by Th28; then
A4: g is_isomorphism_of RelIncl X, RelIncl Y by Th21;
A5: dom g = X by Th18;
    b c/= g.a by A1,Th4; then
    c c/= a by A2,A3,A4,A5,Th6; then
    a in c by Th4; then
    succ a c= c by ORDINAL1:21;
    hence g.succ a c= b by A2,ORDINAL4:9;
  end;
