reserve x for set;
reserve a, b, c for Real;
reserve m, n, m1, m2 for Nat;
reserve k, l for Integer;
reserve p, q for Rational;
reserve s1, s2 for Real_Sequence;

theorem Th18:
  s1 is convergent & (for n holds s2.n = (s1.n) |^ m) implies s2
  is convergent & lim s2 = (lim s1) |^ m
proof
  assume that
A1: s1 is convergent and
A2: for n holds s2.n = (s1.n) |^ m;
  defpred P[Nat] means
for s being Real_Sequence st (for n holds s.
  n=(s1.n) |^ $1) holds s is convergent & lim s=(lim s1) |^ $1;
A3: P[0]
  proof
    let s be Real_Sequence;
    assume
A4: for n holds s.n = (s1.n) |^ 0;
reconsider j = 1 as Element of REAL by NUMBERS:19;

A5: now
      let n be Nat;
      thus s.n = (s1.n) |^ 0 by A4
        .= (s1.n) GeoSeq.0 by Def1
        .= j by Th3;
    end;
    then
A6: s is constant by VALUED_0:def 18;
    hence s is convergent;
    thus lim s = s.0 by A6,SEQ_4:26
      .= 1 by A5
      .= (lim s1) GeoSeq.0 by Th3
      .= (lim s1) |^ 0 by Def1;
  end;
A7: for m1 holds P[m1] implies P[m1+1]
  proof
    let m1;
    assume
A8: for s being Real_Sequence st (for n holds s.n = (s1.n) |^ m1)
    holds s is convergent & lim s = (lim s1) |^ m1;
    deffunc O(Nat) = (s1.$1) |^ m1;
    let s be Real_Sequence;
    consider s3 being Real_Sequence such that
A9: for n holds s3.n = O(n) from SEQ_1:sch 1;
    assume
A10: for n holds s.n = (s1.n) |^ (m1+1);
    now
      let n;
      thus s.n = (s1.n) |^ (m1+1) by A10
        .= (s1.n) |^ m1 * (s1.n) by NEWTON:6
        .= (s3.n) * (s1.n) by A9;
    end;
    then
A11: s = s3(#)s1 by SEQ_1:8;
A12: s3 is convergent by A8,A9;
    hence s is convergent by A1,A11;
    lim s3 = (lim s1) |^ m1 by A8,A9;
    hence lim s = (lim s1) |^ m1 * (lim s1) by A1,A12,A11,SEQ_2:15
      .= (lim s1) |^ (m1+1) by NEWTON:6;
  end;
  for m1 holds P[m1] from NAT_1:sch 2(A3,A7);
  hence thesis by A2;
end;
