reserve P for Subset of TOP-REAL 2,
  f,f1,f2,g for FinSequence of TOP-REAL 2,
  p,p1,p2,q,q1,q2 for Point of TOP-REAL 2,
  r1,r2,r19,r29 for Real,
  i,j,k,n for Nat;

theorem Th3:
  for i, n being Nat holds i+1 <= len(f|n) implies LSeg(f|n,i) = LSeg(f,i)
proof
  let i,n be Nat;
  assume
A1: i+1 <= len(f|n);
  per cases;
  suppose
    i <> 0;
    then
A2: 1 <= i by NAT_1:14;
    then
A3: i in dom(f|n) by A1,SEQ_4:134;
    len(f|n) <= len f by FINSEQ_5:16;
    then
A4: i+1 <= len f by A1,XXREAL_0:2;
A5: i+1 in dom(f|n) by A1,A2,SEQ_4:134;
    thus LSeg(f|n,i) = LSeg((f|n)/.i,(f|n)/.(i+1)) by A1,A2,TOPREAL1:def 3
      .= LSeg(f/.i,(f|n)/.(i+1)) by A3,FINSEQ_4:70
      .= LSeg(f/.i,f/.(i+1)) by A5,FINSEQ_4:70
      .= LSeg(f,i) by A2,A4,TOPREAL1:def 3;
  end;
  suppose
A6: i = 0;
    hence LSeg(f|n,i) = {} by TOPREAL1:def 3
      .= LSeg(f,i) by A6,TOPREAL1:def 3;
  end;
end;
