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 Th24:
  1 in C & A in B implies exp(C,A) in exp(C,B)
proof
  defpred OO[Ordinal] means for A st A in $1 holds exp(C,A) in exp(C,$1);
A1: for B st B <> 0 & B is limit_ordinal & for D st D in B holds OO[D]
  holds OO[B]
  proof
    deffunc F(Ordinal) = exp(C,$1);
    let B such that
A2: B <> 0 and
A3: B is limit_ordinal and
A4: for D st D in B holds OO[D];
    consider fi such that
A5: dom fi = B & for D st D in B holds fi.D = F(D) from ORDINAL2:sch
    3;
    fi is increasing
    proof
      let B1,B2 be Ordinal;
      assume that
A6:   B1 in B2 and
A7:   B2 in dom fi;
A8:   fi.B1 = exp(C,B1) by A5,A6,A7,ORDINAL1:10;
      exp(C,B1) in exp(C,B2) by A4,A5,A6,A7;
      hence thesis by A5,A7,A8;
    end;
    then
A9: sup fi = lim fi by A2,A3,A5,Th8;
    let A such that
A10: A in B;
    fi.A = exp(C,A) by A10,A5;
    then
A11: exp(C,A) in rng fi by A10,A5,FUNCT_1:def 3;
    exp(C,B) = lim fi by A2,A3,A5,ORDINAL2:45;
    hence thesis by A9,A11,ORDINAL2:19;
  end;
  assume
A12: 1 in C;
A13: for B st OO[B] holds OO[succ B]
  proof
    let B such that
A14: for A st A in B holds exp(C,A) in exp(C,B);
    let A;
    assume A in succ B;
    then
A15: A c= B by ORDINAL1:22;
A16: now
      assume A <> B;
      then A c< B by A15;
      hence exp(C,A) in exp(C,B) by A14,ORDINAL1:11;
    end;
    exp(C,B) in exp(C,succ B) by A12,Th23;
    hence thesis by A16,ORDINAL1:10;
  end;
A17: OO[0];
  for B holds OO[B] from ORDINAL2:sch 1(A17,A13,A1);
  hence thesis;
end;
