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 Th20:
  A <> {} implies exp({},A) = {}
proof
  defpred FF[Ordinal] means $1 <> {} implies exp({},$1) = {};
A1: FF[B] implies FF[succ B]
  proof
    assume that
    FF[B] and
    succ B <> {};
    thus exp({},succ B) = {}*^exp({},B) by ORDINAL2:44
      .= {} by ORDINAL2:35;
  end;
A2: for B st B <> 0 & B is limit_ordinal & for C st C in B holds FF[C]
  holds FF[B]
  proof
    deffunc F(Ordinal) = exp({},$1);
    let A such that
A3: A <> 0 and
A4: A is limit_ordinal and
A5: for C st C in A holds FF[C] and
    A <> {};
    consider fi such that
A6: dom fi = A & for B st B in A holds fi.B = F(B) from ORDINAL2:sch 3;
    0 is_limes_of fi
    proof
      per cases;
      case
        0 = 0;
        take B = 1;
        {} in A by A3,ORDINAL3:8;
        hence B in dom fi by A4,A6,Lm3,ORDINAL1:28;
        let D;
        assume
A7:     B c= D;
        assume
A8:     D in dom fi;
        then FF[D] by A5,A6;
        hence thesis by A6,A7,A8,Lm3,ORDINAL1:21;
      end;
      case
        0 <> 0;
        thus thesis;
      end;
    end;
    then lim fi = {} by ORDINAL2:def 10;
    hence thesis by A3,A4,A6,ORDINAL2:45;
  end;
A9: FF[0];
  FF[B] from ORDINAL2:sch 1(A9,A1,A2);
  hence thesis;
end;
