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;

theorem Th11:
  0 in a & b is non empty limit_ordinal implies
  (x in exp(a,b) iff ex c st c in b & x in exp(a,c))
  proof assume
A1: 0 in a & b is non empty limit_ordinal;
    deffunc F(Ordinal) = exp(a,$1);
    consider f such that
A2: dom f = b & for c st c in b holds f.c = F(c) from ORDINAL2:sch 3;
    f is non-decreasing by A1,A2,Th8; then
    Union f is_limes_of f by A1,A2,Th6; then
A3: Union f = lim f by ORDINAL2:def 10 .= exp(a,b) by A1,A2,ORDINAL2:45;
    hereby assume x in exp(a,b); then
      consider c being object such that
A4:   c in dom f & x in f.c by A3,CARD_5:2;
      reconsider c as Ordinal by A4;
      take c;
      thus c in b by A2,A4;
      thus x in exp(a,c) by A2,A4;
    end;
    given c such that
A5: c in b & x in exp(a,c);
    f.c = F(c) by A2,A5;
    hence x in exp(a,b) by A2,A3,A5,CARD_5:2;
  end;
