reserve r1,r2 for Real;
reserve n,i,i1,i2,j for Nat;
reserve D for non empty set;
reserve f for FinSequence of D;

theorem Th4:
  for f being FinSequence of TOP-REAL 2,n being Element of NAT st
  2<=n & f is being_S-Seq holds f|n is being_S-Seq
proof
  let f be FinSequence of TOP-REAL 2,n be Element of NAT;
  assume that
A1: 2<=n and
A2: f is being_S-Seq;
A3: len f >= 2 by A2,TOPREAL1:def 8;
A4: now
    per cases;
    case
      n<=len f;
      hence len (f|n) >= 2 by A1,FINSEQ_1:59;
    end;
    case
      n>len f;
      hence len (f|n) >= 2 by A3,FINSEQ_1:58;
    end;
  end;
  reconsider f9=f as s.n.c. special unfolded one-to-one FinSequence of
  TOP-REAL 2 by A2;
  f9|n is one-to-one;
  hence thesis by A4,TOPREAL1:def 8;
end;
