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 Th10:
  for i being Nat holds p in rng f implies LSeg(f:-p,i+1) = LSeg(f ,i+p..f)
proof
  let i be Nat;
A1: 1 <= i+1 by NAT_1:11;
  assume
A2: p in rng f;
  then
A3: len (f:-p) = len f - p..f + 1 by FINSEQ_5:50;
A4: i+(1+1) = i+1+1;
  per cases;
  suppose
A5: i+2 <= len(f:-p);
    then i+1 <= len f - p..f by A4,A3,XREAL_1:6;
    then
A6: i+1+p..f <= len f by XREAL_1:19;
    1 <= p..f by A2,FINSEQ_4:21;
    then i+1 <= i+p..f by XREAL_1:6;
    then
A7: 1 <= i+p..f by A1,XXREAL_0:2;
A8: i+1 in dom(f:-p) by A1,A4,A5,SEQ_4:134;
A9: i+1+1 in dom(f:-p) by A1,A5,SEQ_4:134;
    thus LSeg(f:-p,i+1) = LSeg((f:-p)/.(i+1),(f:-p)/.(i+1+1)) by A1,A5,
TOPREAL1:def 3
      .= LSeg(f/.(i+p..f),(f:-p)/.(i+1+1)) by A2,A8,FINSEQ_5:52
      .= LSeg(f/.(i+p..f),f/.(i+1+p..f)) by A2,A9,FINSEQ_5:52
      .= LSeg(f/.(i+p..f),f/.(i+p..f+1))
      .= LSeg(f,i+p..f) by A7,A6,TOPREAL1:def 3;
  end;
  suppose
A10: i+2 > len(f:-p);
    then i+1 > len f - p..f by A4,A3,XREAL_1:6;
    then p..f+(i+1) > len f by XREAL_1:19;
    then i+p..f+1 > len f;
    hence LSeg(f,i+p..f) = {} by TOPREAL1:def 3
      .= LSeg(f:-p,i+1) by A4,A10,TOPREAL1:def 3;
  end;
end;
