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 Th16:
  seq^\k is convergent_to_finite_number implies seq is
  convergent_to_finite_number & seq is convergent & lim seq = lim(seq^\k)
proof
  set seq0=seq^\k;
  assume
A1: seq^\k is convergent_to_finite_number;
  then
A2: not(lim seq0 = +infty & seq0 is convergent_to_+infty) by MESFUNC5:50;
A3: not(lim seq0 = -infty & seq0 is convergent_to_-infty) by A1,MESFUNC5:51;
  seq0 is convergent by A1,MESFUNC5:def 11;
  then
A4: ex g be Real st lim seq0 = g & (for p be Real st 0<p ex n
  be Nat st for m be Nat st n<=m holds |.seq0.m-lim seq0 .| < p) & seq0 is
  convergent_to_finite_number by A2,A3,MESFUNC5:def 12;
  then consider g be Real such that
A5: lim seq0 = g;
A6: for p be Real st 0<p ex n be Nat st for m be Nat st n<=m holds |.
  seq.m-lim seq0 .| < 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 |.seq0.m-lim seq0 .| < p by A4;
    take n1 = n+k;
    hereby
      let m be Nat;
      assume
A8:   n1 <= m;
      k <= n+k by NAT_1:11;
      then reconsider mk = m-k as Element of NAT by A8,INT_1:5,XXREAL_0:2;
A9:   seq0.(m-k) = seq.(mk+k) by NAT_1:def 3;
      n+k-k <= m-k by A8,XREAL_1:9;
      hence |.seq.m -lim seq0 .| < p by A7,A9;
    end;
  end;
  lim seq0 = g by A5;
  hence
A10: seq is convergent_to_finite_number by A6,MESFUNC5:def 8;
  hence seq is convergent by MESFUNC5:def 11;
  hence thesis by A5,A6,A10,MESFUNC5:def 12;
end;
