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 Th47:
  a in dom g & b in dom criticals g
  implies (criticals g).b is_a_fixpoint_of g.a
  proof
    assume that
A1: a in dom g and
A2: b in dom criticals g;
    set h = criticals g;
    set X = {c where c is Element of dom(g.0): c in dom(g.0) &
    for f st f in rng g holds c is_a_fixpoint_of f};
    X is ordinal-membered by Th46; then
    rng h = X by Th19; then
    h.b in X by A2,FUNCT_1:def 3; then
    consider c being Element of dom(g.0) such that
A3: h.b = c & c in dom(g.0) & for f st f in rng g holds c is_a_fixpoint_of f;
    g.a in rng g by A1,FUNCT_1:def 3;
    hence (criticals g).b is_a_fixpoint_of g.a by A3;
  end;
