reserve phi,fi,psi for Ordinal-Sequence,
  A,A1,B,C,D for Ordinal,
  f,g for Function,
  X for set,
  x,y,z for object;
reserve f1,f2 for Ordinal-Sequence;

theorem Th22:
  C <> {} implies exp(C,A) <> {}
proof
  defpred P[Ordinal] means exp(C,$1) <> {};
  assume
A1: C <> {};
A2: for A st P[A] holds P[succ A]
  proof
    let A such that
A3: exp(C,A) <> {};
    exp(C,succ A) = C*^exp(C,A) by ORDINAL2:44;
    hence thesis by A1,A3,ORDINAL3:31;
  end;
A4: for A st A <> 0 & A is limit_ordinal & for B st B in A holds P[B] holds
  P[A]
  proof
    let A such that
A5: A <> 0 and
A6: A is limit_ordinal and
A7: for B st B in A holds exp(C,B) <> {};
    consider fi such that
A8: dom fi = A and
A9: for B st B in A holds fi.B = exp(C,B) and
A10: ex D st D is_limes_of fi by A5,A6,Lm8;
A11: exp(C,A) = lim fi by A5,A6,A8,A9,ORDINAL2:45;
    assume
A12: exp(C,A) = {};
    consider D such that
A13: D is_limes_of fi by A10;
    lim fi = D by A13,ORDINAL2:def 10;
    then consider B such that
A14: B in dom fi and
A15: for D st B c= D & D in dom fi holds fi.D = {} by A11,A13,A12,
ORDINAL2:def 9;
    fi.B = exp(C,B) by A8,A9,A14;
    hence contradiction by A7,A8,A14,A15;
  end;
A16: P[0] by ORDINAL2:43;
  for A holds P[A] from ORDINAL2:sch 1(A16,A2,A4);
  hence thesis;
end;
