reserve i,j,k,n,m for Nat;
reserve p,q for Point of TOP-REAL 2;
reserve G for Go-board;
reserve C for Subset of TOP-REAL 2;

theorem
  for f, g being FinSequence of TOP-REAL 2 st g is_in_the_area_of f
  holds Rev g is_in_the_area_of f
proof
  let f, g be FinSequence of TOP-REAL 2 such that
A1: g is_in_the_area_of f;
A2: Rev Rev g = g;
  let n such that
A3: n in dom Rev g;
  n >= 1 by A3,FINSEQ_3:25;
  then
A4: n-1 >= 0 by XREAL_1:48;
  set i = len g + 1 -' n;
A5: len Rev g = len g by FINSEQ_5:def 3;
  then
A6: n <= len g by A3,FINSEQ_3:25;
  then
A7: i = len g -' n + 1 by NAT_D:38;
  then i = len g - n + 1 by A6,XREAL_1:233
    .= len g - (n-1);
  then
A8: i <= len g - 0 by A4,XREAL_1:13;
  1 <= i by A7,NAT_1:11;
  then
A9: i in dom g by A8,FINSEQ_3:25;
  len g <= len g + 1 by NAT_1:11;
  then n + i = len g + 1 by A6,XREAL_1:235,XXREAL_0:2;
  then (Rev g)/.n = g/.i by A2,A5,A3,FINSEQ_5:66;
  hence thesis by A1,A9;
end;
