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 Th65:
  0 in Segm n implies <%n*^exp(omega,b)%> is Cantor-normal-form
proof assume
A1: 0 in Segm n;
    set A = <%n*^exp(omega,b)%>;
A2: dom A = Segm 1 & A.0 = n*^exp(omega,b) by AFINSQ_1:def 4;
    hereby let a; assume a in dom A; then
      a = 0 & 0 in Segm 1 by A2,ORDINAL3:15,TARSKI:def 1;
      hence A.a is Cantor-component by A1;
    end;
    let a,b; assume
A3: a in b;
    assume b in dom A;
    hence omega-exponent(A.b) in omega-exponent(A.a)
      by A3,A2,ORDINAL3:15,TARSKI:def 1;
  end;
