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 Th69:
  b-Veblen((succ b)-Veblen a) = (succ b)-Veblen a
  proof set U = Tarski-Class(b\/a\/omega);
A1: omega in U by Th57;
    reconsider b1 = b as Ordinal of U by Th66;
    succ b1 in On U by ORDINAL1:def 9; then
A2: U-Veblen.(succ b) = criticals (U-Veblen.b) by Def15;
    reconsider f = U-Veblen.b1, g = U-Veblen.(succ b1) as
    normal Ordinal-Sequence of U by A1,Th62;
A3: a in U by Th66; then
A4: a in On U by ORDINAL1:def 9;
A5: dom f = On U & dom g = On U by FUNCT_2:def 1;
    set W = Tarski-Class(b\/(g.a)\/omega);
    omega in U by Th57;
    then
A6: (succ b1)-Veblen a = g.a & b1-Veblen(g.a) = f.(g.a) by A3,Th67; then
    (succ b)-Veblen a is_a_fixpoint_of U-Veblen.b by A4,A2,A5,Th29;
    hence b-Veblen((succ b)-Veblen a) = (succ b)-Veblen a by A6;
  end;
