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 Th60:
  a c= b & b in U & omega in U & c in dom(U-Veblen.b) &
  (for c st c in b holds U-Veblen.c is normal)
  implies U-Veblen.a.c c= U-Veblen.b.c
  proof assume
A1: a c= b & b in U & omega in U & c in dom(U-Veblen.b);
    set F = U-Veblen;
    defpred P[Ordinal] means
    for a,c st a c= $1 & $1 in U & c in dom(F.$1) &
    for c st c in $1 holds U-Veblen.c is normal holds F.a.c c= F.$1.c;
A2: P[0];
A3: for b st P[b] holds P[succ b]
    proof
      let b such that
A4:   P[b];
      let a,c such that
A5:   a c= succ b and
A6:   succ b in U and
A7:   c in dom(F.succ b);
      assume
A8:   for c st c in succ b holds U-Veblen.c is normal;
      succ b in On U by A6,ORDINAL1:def 9; then
A9:   F.succ b = criticals (F.b) by Def15; then
A10:   dom(F.succ b) c= dom(F.b) by Th32;
A11:   b in succ b by ORDINAL1:6; then
A12:   b in U by A6,ORDINAL1:10;
      F.b is normal by A8,ORDINAL1:6; then
A13:   F.b.c c= F.succ b.c by A7,A9,Th45;
A14:   for c st c in b holds F.c is normal by A8,A11,ORDINAL1:10;
      per cases by A5,ORDINAL5:1;
      suppose a = succ b;
        hence thesis;
      end;
      suppose a c= b; then
        F.a.c c= F.b.c by A4,A7,A10,A12,A14;
        hence thesis by A13;
      end;
    end;
A15: for b st b <> 0 & b is limit_ordinal & for d st d in b holds P[d]
    holds P[b]
    proof
      let b;
      assume
A16:   b <> 0 & b is limit_ordinal;
      assume for d st d in b holds P[d];
      let a,c;
      assume
A17:   a c= b;
      per cases by A17;
      suppose a = b;
        hence thesis;
      end;
      suppose
A18:     a c< b; then
A19:     a in b by ORDINAL1:11;
        assume
        b in U; then
A20:     b in On U by ORDINAL1:def 9; then
A21:     F.b = criticals(F|b) by A16,Def15;
        dom F = On U by Def15; then
        b c= dom F by A20,ORDINAL1:def 2; then
A22:     dom(F|b) = b by RELAT_1:62;
        assume
A23:     c in dom(F.b);
       assume
A24:     for c st c in b holds U-Veblen.c is normal;
A25:     now let c; assume c in dom(F|b); then
          c in b & (F|b).c = F.c by A22,FUNCT_1:49;
          hence (F|b).c is normal by A24;
        end;
A26:     (F|b).a in rng(F|b) by A19,A22,FUNCT_1:def 3;
        (F|b).a = F.a by A18,FUNCT_1:49,ORDINAL1:11;
        hence F.a.c c= F.b.c by A19,A21,A22,A23,A25,A26,Th54;
      end;
    end;

    for b holds P[b] from ORDINAL2:sch 1(A2,A3,A15);
    hence thesis by A1;
  end;
