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;
reserve U for uncountable Universe;

theorem Th76:
  for e being epsilon Ordinal ex a st e = 1-Veblen a
  proof
    let e be epsilon Ordinal;
    set U = Tarski-Class(e\/omega);
A1: omega in U by Th57;
    0-element_of U = 0 & 1-element_of U = 1; then
    reconsider f = U-Veblen.0, g = U-Veblen.1 as
    normal Ordinal-Sequence of U by A1,Th62;
A2: g = criticals(U exp omega) by Th74
    .= criticals f by Def15;
A3: f.e = 0-Veblen e .= exp(omega,e) by Th68 .= e by ORDINAL5:def 5;
A4: dom f = On U by FUNCT_2:def 1;
    e c= e\/omega & e\/omega in U by CLASSES1:2,XBOOLE_1:7; then
A5: e in U by CLASSES1:def 1; then
    e in On U by ORDINAL1:def 9; then
    e is_a_fixpoint_of f by A3,A4; then
    consider a such that
A6: a in dom criticals f & e = (criticals f).a by Th33;
    take a;
    set W = Tarski-Class(a\/omega);
A7: a c= a\/omega & a\/omega in W by CLASSES1:2,XBOOLE_1:7;
    a c= e by A6,ORDINAL4:10; then
    omega in W & a in W & a in U & 1-element_of U = 1-element_of W
    by A5,A7,Th57,CLASSES1:def 1;
    hence e = 1-Veblen a by A1,A2,A6,Th64;
  end;
