reserve
  a,b,c,d,e for Ordinal,
  m,n for Nat,
  f for Ordinal-Sequence,
  x for object;
reserve S,S1,S2 for Sequence;
reserve A,B for Cantor-normal-form Ordinal-Sequence;

theorem Th66:
  for A,B being Ordinal-Sequence st A^B is Cantor-normal-form
  holds A is Cantor-normal-form & B is Cantor-normal-form
  proof
    let A,B be Ordinal-Sequence;
    set AB = A^B;
    assume that
A1: for a st a in dom (A^B) holds (A^B).a is Cantor-component and
A2: for a,b st a in b & b in dom (A^B) holds
    omega-exponent((A^B).b) in omega-exponent((A^B).a);
A3: dom (A^B) = (dom A)+^(dom B) by ORDINAL4:def 1; then
A4: dom A c= dom (A^B) by ORDINAL3:24;
    thus A is Cantor-normal-form
    proof
      hereby let a; assume a in dom A; then
        A.a = (A^B).a & a in dom (A^B) by A4,ORDINAL4:def 1;
        hence A.a is Cantor-component by A1;
      end;
      let a,b; assume
A5:   a in b & b in dom A; then
      a in dom A by ORDINAL1:10; then
      A.a = AB.a & A.b = AB.b & b in dom AB by A4,A5,ORDINAL4:def 1;
      hence omega-exponent(A.b) in omega-exponent(A.a) by A2,A5;
    end;
    hereby let a; assume a in dom B; then
      B.a = AB.((dom A)+^a) & (dom A)+^a in dom AB
      by A3,ORDINAL3:17,ORDINAL4:def 1;
      hence B.a is Cantor-component by A1;
    end;
    let a,b; assume
A6: a in b & b in dom B; then
    a in dom B by ORDINAL1:10; then
    B.a = AB.((dom A)+^a) & B.b = AB.((dom A)+^b) & (dom A)+^b in dom AB &
    (dom A)+^a in (dom A)+^b by A3,A6,ORDINAL3:17,ORDINAL4:def 1;
    hence omega-exponent(B.b) in omega-exponent(B.a) by A2;
  end;
