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 Th63:
  omega in U & U c= W & a in U implies U-Veblen.a c= W-Veblen.a
  proof assume
A1: omega in U & U c= W; then
A2: On U c= On W by ORDINAL2:9;
    defpred P[Ordinal] means
    $1 in U implies U-Veblen.$1 c= W-Veblen.$1;
A3: U-Veblen.0 = U exp omega & W-Veblen.0 = W exp omega by Def15;
A4: dom (U exp omega) = On U & dom (W exp omega) = On W by FUNCT_2:def 1;
    now let x be object; assume
      x in On U; then
      reconsider a = x as Ordinal of U by ORDINAL1:def 9;
      reconsider b = a as Ordinal of W by A1;
      (U exp omega).a = exp(omega,b) by A1,Def8;
      hence (U exp omega).x = (W exp omega).x by A1,Def8;
    end; then
A5: P[0] by A2,A3,A4,GRFUNC_1:2;
A6: P[b] implies P[succ b]
    proof assume
A7:   P[b]; assume
A8:   succ b in U;
A9:   b in succ b by ORDINAL1:6;
      succ b in On U & succ b in On W by A1,A8,ORDINAL1:def 9; then
      U-Veblen.succ b = criticals (U-Veblen.b) &
      W-Veblen.succ b = criticals (W-Veblen.b) by Def15;
      hence thesis by A7,A9,Th40,A8,ORDINAL1:10;
    end;
A10: b <> 0 & b is limit_ordinal & (for c st c in b holds P[c]) implies P[b]
    proof assume that
A11:   b <> 0 & b is limit_ordinal and
A12:   for c st c in b holds P[c] and
A13:   b in U;
      set g1 = U-Veblen|b, g2 = W-Veblen|b;
A14:   b in On U & b in On W by A1,A13,ORDINAL1:def 9; then
A15:   U-Veblen.b = criticals g1 & W-Veblen.b = criticals g2 by A11,Def15;
      dom(U-Veblen) = On U & dom(W-Veblen) = On W by Def15; then
      b c= dom(U-Veblen) & b c= dom(W-Veblen) by A14,ORDINAL1:def 2; then
A16:   dom g1 = b & dom g2 = b by RELAT_1:62;
      now
        let a; assume
A17:     a in dom g1; then
A18:     g1.a = U-Veblen.a & g2.a = W-Veblen.a by A16,FUNCT_1:47;
        a in U by A13,A16,A17,ORDINAL1:10;
        hence g1.a c= g2.a by A12,A16,A17,A18;
      end;
      hence thesis by A11,A15,A16,Th55;
    end;
    P[b] from ORDINAL2:sch 1(A5,A6,A10);
    hence thesis;
  end;
