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 Th64:
  omega in U & a in U & b in U & omega in W & a in W & b in W implies
  U-Veblen.b.a = W-Veblen.b.a
  proof assume
A1: omega in U & a in U & b in U & omega in W & a in W & b in W; then
A2: a in On U & a in On W by ORDINAL1:def 9;
    W-Veblen.b is Ordinal-Sequence of W &
    U-Veblen.b is Ordinal-Sequence of U by A1,Th62; then
A3: dom(U-Veblen.b) = On U & dom(W-Veblen.b) = On W by FUNCT_2:def 1;
    U c= W or W in U by CLASSES2:53; then
    U c= W or W c= U by ORDINAL1:def 2; then
    U-Veblen.b c= W-Veblen.b or W-Veblen.b c= U-Veblen.b by A1,Th63;
    hence thesis by A2,A3,GRFUNC_1:2;
  end;
