reserve r,r1,g for Real,
  n,m,k for Nat,
  seq,seq1, seq2 for Real_Sequence,
  f,f1,f2 for PartFunc of REAL,REAL,
  x for set;

theorem Th5:
  seq is convergent & 0<lim seq implies ex n st for m st n<=m
  holds (lim seq)/2<seq.m
proof
  assume that
A1: seq is convergent and
A2: 0<lim seq;
  reconsider ls = (lim seq)/2 as Element of REAL by XREAL_0:def 1;
  set s1 = seq_const((lim seq)/2);
A3: seq-s1 is convergent by A1;
  s1.0=(lim seq)/2 by SEQ_1:57;
  then lim(seq-s1)=(lim seq)/2+(lim seq)/2-(lim seq)/2 by A1,SEQ_4:42
    .=(lim seq)/2;
  then consider n such that
A4: for m st n<=m holds 0<(seq-s1).m by A2,A3,Th4,XREAL_1:215;
  take n;
  let m;
  assume n<=m;
  then 0<(seq-s1).m by A4;
  then 0<seq.m-s1.m by RFUNCT_2:1;
  then 0<seq.m-(lim seq)/2 by SEQ_1:57;
  then 0+(lim seq)/2<seq.m by XREAL_1:20;
  hence thesis;
end;
