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 Th70:
  b in c implies b-Veblen(c-Veblen a) = c-Veblen a
  proof assume
A1: b in c;
    set U = Tarski-Class(c\/a\/omega);
A2: omega in U by Th57;
A3: a in U & c in U by Th66; then
A4: b in U by A1,ORDINAL1:10; then
    reconsider f = U-Veblen.b, g = U-Veblen.c as normal Ordinal-Sequence of U
    by A2,Th66,Th62;
    dom g = On U by FUNCT_2:def 1; then
    a in dom g by A3,ORDINAL1:def 9; then
    g.a is_a_fixpoint_of f by A1,A2,Th66,Th58; then
    g.a = f.(g.a);
    hence b-Veblen(c-Veblen a) = c-Veblen a by A2,A4,Th67;
  end;
