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 Th68:
  for A being non empty Cantor-normal-form Ordinal-Sequence
  for c being Cantor-component Ordinal
  st omega-exponent(A.0) in omega-exponent(c)
  holds <%c%>^A is Cantor-normal-form
  proof
    let A be non empty Cantor-normal-form Ordinal-Sequence;
    let c be Cantor-component Ordinal such that
A1: omega-exponent(A.0) in omega-exponent(c);
    set B = <%c%>^A;
A2: dom <%c%> = 1 & <%c%>.0 = c by AFINSQ_1:def 4; then
A3: dom B = 1 +^ dom A by ORDINAL4:def 1;
    hereby let a;
      assume
A4:   a in dom B;
      per cases by A3,A4,ORDINAL3:38;
      suppose
A5:     a in 1; then
        a = 0 by ORDINAL3:15,TARSKI:def 1;
        hence B.a is Cantor-component by A2,A5,ORDINAL4:def 1;
      end;
      suppose ex b st b in dom A & a = 1 +^b; then
        consider b such that
A6:     b in dom A & a = 1 +^b;
        B.a = A.b by A2,A6,ORDINAL4:def 1;
        hence B.a is Cantor-component by A6,Def11;
      end;
    end;
    let a,b; assume
A7: a in b & b in dom B;
    per cases;
    suppose not a in 1; then
A8:   1 c= a by ORDINAL1:16; then
A9:   1 in b & a in dom B & (1+^dom A)-^1 = dom A
      by A7,ORDINAL1:10,12,ORDINAL3:52; then
A10:   b-^1 in dom A & a-^1 in dom A & a-^1 in b-^1 by A8,A3,A7,ORDINAL3:53;
      b = 1+^(b-^1) & a = 1+^(a-^1) by A8,A9,ORDINAL3:51,def 5; then
      B.a = A.(a-^1) & B.b = A.(b-^1) by A2,A10,ORDINAL4:def 1;
      hence thesis by A10,Def11;
    end;
    suppose a in 1; then
A11:   B.a = <%c%>.a & a = 0 by A2,ORDINAL3:15,ORDINAL4:def 1,TARSKI:def 1;
 then
A12:   succ 0 c= b by A7,ORDINAL1:21; then
      b-^1 in (1+^dom A)-^1 by A3,A7,ORDINAL3:53; then
A13:   b-^1 in dom A & b = 1+^(b-^1) by A12,ORDINAL3:52,def 5; then
A14:   B.b = A.(b-^1) by A2,ORDINAL4:def 1;
      0 in dom A & (b-^1 = 0 or 0 in b-^1) by ORDINAL3:8; then
      omega-exponent(A.0) = omega-exponent(A.(b-^1)) or
      omega-exponent(A.(b-^1)) in omega-exponent(A.0) by A13,Def11;
      hence thesis by A1,A11,A14,ORDINAL1:10;
    end;
  end;
