reserve n for Nat;

theorem Th2:
  for f,g be FinSequence of TOP-REAL 2 st f is_in_the_area_of g for
  p be Element of TOP-REAL 2 st p in rng f holds f:-p is_in_the_area_of g
proof
  let f,g be FinSequence of TOP-REAL 2;
  assume
A1: f is_in_the_area_of g;
  let p be Element of TOP-REAL 2;
  assume
A2: p in rng f;
  let n be Nat;
  1 <= p..f by A2,FINSEQ_4:21;
  then
A3: 0+1 <= n-'1+p..f by XREAL_1:7;
  assume
A4: n in dom(f:-p);
  then
A5: n in Seg len(f:-p) by FINSEQ_1:def 3;
  then
A6: 0+1 <= n by FINSEQ_1:1;
  then n-'1+1 = n by XREAL_1:235;
  then
A7: (f:-p)/.n = f/.(n-'1+p..f) by A2,A4,FINSEQ_5:52;
  len(f:-p) = len f - p..f + 1 by A2,FINSEQ_5:50;
  then n <= len f - p..f + 1 by A5,FINSEQ_1:1;
  then n - 1 <= len f - p..f by XREAL_1:20;
  then
A8: n - 1 + p..f <= len f by XREAL_1:19;
  n-1 >= 0 by A6,XREAL_1:19;
  then n-'1+p..f <= len f by A8,XREAL_0:def 2;
  then n-'1+p..f in dom f by A3,FINSEQ_3:25;
  hence thesis by A1,A7;
end;
