reserve X for non empty set,
        x for Element of X,
        S for SigmaField of X,
        M for sigma_Measure of S,
        f,g,f1,g1 for PartFunc of X,REAL,
        l,m,n,n1,n2 for Nat,
        a,b,c for Real;
reserve k for positive Real;

theorem Th9:
for seq1,seq2 be Real_Sequence, k be positive Real st
  for n be Nat holds seq1.n = (seq2.n) to_power k & seq2.n >= 0
holds (seq1 is convergent iff seq2 is convergent)
proof
   let seq1,seq2 be Real_Sequence, k be positive Real;
   assume
A1: for n be Nat holds seq1.n = (seq2.n) to_power k & seq2.n >= 0;
A2:for n holds seq1.n >= 0
   proof
    let n;
    (seq2.n) to_power k >= 0 by A1,Th4;
    hence thesis by A1;
   end;
   thus seq1 is convergent implies seq2 is convergent
   proof
    assume A3: seq1 is convergent;
    for n be Nat holds seq2.n = (seq1.n) to_power (1/k)
    proof
     let n be Nat;
     (seq1.n) to_power (1/k) = (seq2.n) to_power k to_power (1/k) by A1
      .= (seq2.n) to_power (k*(1/k)) by A1,HOLDER_1:2
      .= (seq2.n) to_power 1 by XCMPLX_1:106;
     hence thesis by POWER:25;
    end;
    hence thesis by A2,A3,HOLDER_1:10;
   end;
   assume seq2 is convergent;
   hence thesis by A1,HOLDER_1:10;
end;
