reserve i, i1, i2, j, k for Nat,
  r, s for Real;
reserve D for non empty set,
  f1 for FinSequence of D;

theorem Th7:
  for g being FinSequence of TOP-REAL 2,i st g is being_S-Seq &
  i+1<len g holds g/^i is being_S-Seq
proof
  let g be FinSequence of TOP-REAL 2,i;
  assume that
A1: g is being_S-Seq and
A2: i+1<len g;
A3: mid(g,i+1,len g)=g/^(i+1-'1) by A2,FINSEQ_6:117
    .=g/^i by NAT_D:34;
A4: 1<=i+1 by NAT_1:11;
  then 1<len g by A2,XXREAL_0:2;
  hence thesis by A1,A2,A4,A3,JORDAN3:6;
end;
