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 Th62:
  omega in U & a in U implies U-Veblen.a is normal Ordinal-Sequence of U
  proof assume
A1: omega in U;
    defpred P[Ordinal] means
    $1 in U implies U-Veblen.$1 is normal Ordinal-Sequence of U;
A2: P[0] by A1,Lm1;
A3: P[b] implies P[succ b]
    proof
      b in succ b by ORDINAL1:6;
      then succ b in U implies b in U by ORDINAL1:10;
      hence thesis by A1,Lm2;
    end;
A4: b <> 0 & b is limit_ordinal & (for c st c in b holds P[c]) implies P[b]
    proof assume that
A5:   b <> 0 & b is limit_ordinal and
A6:   for c st c in b holds P[c] and
A7:   b in U;
      now
        let a; assume
A8:     a in b;
        then a in U by A7,ORDINAL1:10;
        hence U-Veblen.a is normal Ordinal-Sequence of U by A6,A8;
      end;
      hence thesis by A5,A7,Lm3;
    end;
    P[b] from ORDINAL2:sch 1(A2,A3,A4);
    hence thesis;
  end;
