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 Th4:
  for i, n being Nat holds n <= len f & 1 <= i implies LSeg(f/^n,i)
  = LSeg(f,n+i)
proof
  let i,n be Nat;
  assume that
A1: n <= len f and
A2: 1 <= i;
  per cases;
  suppose
A3: i+1 <= len f - n;
    1<=i+1 by NAT_1:11;
    then 1 <= len f - n by A3,XXREAL_0:2;
    then
A4: 1+n <= len f by XREAL_1:19;
    n <= 1+n by NAT_1:11;
    then n <= len f by A4,XXREAL_0:2;
    then
A5: len(f/^n) = len f - n by RFINSEQ:def 1;
    then
A6: i in dom(f/^n) by A2,A3,SEQ_4:134;
A7: i+1+n <= len f by A3,XREAL_1:19;
    i <= i+n by NAT_1:11;
    then
A8: 1 <= i+n by A2,XXREAL_0:2;
A9: i+1 in dom(f/^n) by A2,A3,A5,SEQ_4:134;
    thus LSeg(f/^n,i) = LSeg((f/^n)/.i,(f/^n)/.(i+1)) by A2,A3,A5,
TOPREAL1:def 3
      .= LSeg(f/.(i+n),(f/^n)/.(i+1)) by A6,FINSEQ_5:27
      .= LSeg(f/.(i+n),f/.(i+1+n)) by A9,FINSEQ_5:27
      .= LSeg(f/.(i+n),f/.(i+n+1))
      .= LSeg(f,n+i) by A8,A7,TOPREAL1:def 3;
  end;
  suppose
A10: i+1 > len f - n;
    then n+(i+1) > len f by XREAL_1:19;
    then
A11: n+i+1 > len f;
    i+1 > len(f/^n) by A1,A10,RFINSEQ:def 1;
    hence LSeg(f/^n,i) = {} by TOPREAL1:def 3
      .= LSeg(f,n+i) by A11,TOPREAL1:def 3;
  end;
end;
