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 Th6:
  for i being Nat holds i+1 <= len f implies LSeg(f^g,i) = LSeg(f,i )
proof
  let i be Nat;
  assume
A1: i+1 <= len f;
  per cases;
  suppose
    i <> 0;
    then
A2: 1 <= i by NAT_1:14;
    then
A3: i in dom f by A1,SEQ_4:134;
    len(f^g) = len f + len g by FINSEQ_1:22;
    then len(f^g) >= len f by NAT_1:11;
    then
A4: i+1 <= len(f^g) by A1,XXREAL_0:2;
A5: i+1 in dom f by A1,A2,SEQ_4:134;
    thus LSeg(f,i) = LSeg(f/.i,f/.(i+1)) by A1,A2,TOPREAL1:def 3
      .= LSeg((f^g)/.i,f/.(i+1)) by A3,FINSEQ_4:68
      .= LSeg((f^g)/.i,(f^g)/.(i+1)) by A5,FINSEQ_4:68
      .= LSeg(f^g,i) by A2,A4,TOPREAL1:def 3;
  end;
  suppose
A6: i = 0;
    hence LSeg(f^g,i) = {} by TOPREAL1:def 3
      .= LSeg(f,i) by A6,TOPREAL1:def 3;
  end;
end;
