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
  1 in C implies A c= exp(C,A)
proof
  defpred P[Ordinal] means $1 c= exp(C,$1);
  assume
A1: 1 in C;
A2: P[B] implies P[succ B]
  proof
    assume
A3: B c= exp(C,B);
A4: exp(C,B) = 1*^exp(C,B) by ORDINAL2:39;
    exp(C,succ B) = C*^exp(C,B) by ORDINAL2:44;
    then exp(C,B) in exp(C,succ B) by A1,A4,Th22,ORDINAL2:40;
    then B in exp(C,succ B) by A3,ORDINAL1:12;
    hence thesis by ORDINAL1:21;
  end;
A5: for A st A <> 0 & A is limit_ordinal & for B st B in A holds P[B] holds
  P[A]
  proof
    deffunc F(Ordinal) = exp(C,$1);
    let A such that
A6: A <> 0 and
A7: A is limit_ordinal and
A8: for B st B in A holds B c= exp(C,B);
    consider fi such that
A9: dom fi = A & for B st B in A holds fi.B = F(B) from ORDINAL2:sch
    3;
    let x be object;
    assume
A10: x in A;
    then reconsider B = x as Ordinal;
    fi.B = exp(C,B) by A9,A10;
    then exp(C,B) in rng fi by A9,A10,FUNCT_1:def 3;
    then
A11: exp(C,B) in sup fi by ORDINAL2:19;
    fi is increasing by A1,A9,Th25;
    then
A12: sup fi = lim fi by A6,A7,A9,Th8
      .= exp(C,A) by A6,A7,A9,ORDINAL2:45;
    B c= exp(C,B) by A8,A10;
    hence thesis by A12,A11,ORDINAL1:12;
  end;
A13: P[0] by XBOOLE_1:2;
  P[B] from ORDINAL2:sch 1(A13,A2,A5);
  hence thesis;
end;
