reserve n,m,k,k1,k2 for Nat;
reserve X for non empty Subset of ExtREAL;
reserve Y for non empty Subset of REAL;
reserve seq for ExtREAL_sequence;
reserve e1,e2 for ExtReal;
reserve rseq for Real_Sequence;

theorem Th20:
  seq is convergent_to_finite_number implies seq^\k is
  convergent_to_finite_number & seq^\k is convergent & lim seq = lim(seq^\k)
proof
  set seq0=seq^\k;
  assume
A1: seq is convergent_to_finite_number;
  then
A2: not (lim seq = +infty & seq is convergent_to_+infty) by MESFUNC5:50;
A3: not (lim seq = -infty & seq is convergent_to_-infty) by A1,MESFUNC5:51;
  seq is convergent by A1,MESFUNC5:def 11;
  then
A4: ex g be Real st lim seq = g & (for p be Real st 0<p ex n
  be Nat st for m be Nat st n<=m holds |.seq.m-lim seq .| < p ) & seq is
  convergent_to_finite_number by A2,A3,MESFUNC5:def 12;
  then consider g be Real such that
A5: lim seq = g;
A6: for p be Real st 0<p ex n be Nat st for m be Nat st n<=m holds |.
  seq0.m-lim seq .| < p
  proof
    let p be Real;
    assume 0<p;
    then consider n be Nat such that
A7: for m be Nat st n<=m holds |.seq.m-lim seq .| < p by A4;
    take n;
    hereby
      let m be Nat such that
A8:   n <= m;
      m <= m+k by NAT_1:11;
      then n <= m+k by A8,XXREAL_0:2;
      then |.seq.(m+k) -lim seq .| < p by A7;
      hence |.seq0.m -lim seq .| < p by NAT_1:def 3;
    end;
  end;
  lim seq = g by A5;
  hence
A9: seq0 is convergent_to_finite_number by A6,MESFUNC5:def 8;
  hence seq0 is convergent by MESFUNC5:def 11;
  hence thesis by A5,A6,A9,MESFUNC5:def 12;
end;
