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 Th35:
  for f being rectangular special_circular_sequence holds L~f = {
p where p is Point of TOP-REAL 2: p`1 = W-bound L~f & p`2 <= N-bound L~f & p`2
  >= S-bound L~f or p`1 <= E-bound L~f & p`1 >= W-bound L~f & p`2 = N-bound L~f
or p`1 <= E-bound L~f & p`1 >= W-bound L~f & p`2 = S-bound L~f or p`1 = E-bound
  L~f & p`2 <= N-bound L~f & p`2 >= S-bound L~f}
proof
  let f be rectangular special_circular_sequence;
  set C = L~f, E = { p : p`1 = E-bound C & p`2 <= N-bound C & p`2 >= S-bound C
}, S = { p : p`1 <= E-bound C & p`1 >= W-bound C & p`2 = S-bound C}, N = { p :
p`1 <= E-bound C & p`1 >= W-bound C & p`2 = N-bound C}, W = { p : p`1 = W-bound
  C & p`2 <= N-bound C & p`2 >= S-bound C};
A1: LSeg(SE-corner C, NE-corner C) = E by SPRECT_1:23;
A2: LSeg(SW-corner C, SE-corner C) = S by SPRECT_1:24;
A3: LSeg(SW-corner C, NW-corner C) = W by SPRECT_1:26;
A4: LSeg(NW-corner C, NE-corner C) = N by SPRECT_1:25;
A5: C = L~SpStSeq C by Th34;
  thus C c= { p where p is Point of TOP-REAL 2: p`1 = W-bound C & p`2 <=
  N-bound C & p`2 >= S-bound C or p`1 <= E-bound C & p`1 >= W-bound C & p`2 =
  N-bound C or p`1 <= E-bound C & p`1 >= W-bound C & p`2 = S-bound C or p`1 =
  E-bound C & p`2 <= N-bound C & p`2 >= S-bound C}
  proof
    let x be object;
    assume
A6: x in C;
    then reconsider q=x as Point of TOP-REAL 2;
    q in (LSeg(NW-corner C,NE-corner C) \/ LSeg(NE-corner C,SE-corner C))
\/ (LSeg(SE-corner C,SW-corner C) \/ LSeg(SW-corner C,NW-corner C)) by A5,A6,
SPRECT_1:41;
    then
    q in (LSeg(NW-corner C,NE-corner C) \/ LSeg(NE-corner C,SE-corner C))
    or q in (LSeg(SE-corner C,SW-corner C) \/ LSeg(SW-corner C,NW-corner C))
by XBOOLE_0:def 3;
    then
    q in LSeg(NW-corner C,NE-corner C) or q in LSeg(NE-corner C,SE-corner
C) or q in LSeg(SE-corner C,SW-corner C) or q in LSeg(SW-corner C,NW-corner C)
    by XBOOLE_0:def 3;
    then (ex p st x = p & p`1 = E-bound C & p`2 <= N-bound C & p`2 >= S-bound
C) or (ex p st x = p & p`1 <= E-bound C & p`1 >= W-bound C & p`2 = S-bound C)
or (ex p st x = p & p`1 <= E-bound C & p`1 >= W-bound C & p`2 = N-bound C) or
ex p st x = p & p`1 = W-bound C & p`2 <= N-bound C & p`2 >= S-bound C by A1,A2
,A4,A3;
    hence thesis;
  end;
  let x be object;
  assume x in { p where p is Point of TOP-REAL 2: p`1 = W-bound C & p`2 <=
  N-bound C & p`2 >= S-bound C or p`1 <= E-bound C & p`1 >= W-bound C & p`2 =
  N-bound C or p`1 <= E-bound C & p`1 >= W-bound C & p`2 = S-bound C or p`1 =
  E-bound C & p`2 <= N-bound C & p`2 >= S-bound C};
  then ex p st x = p & (p`1 = W-bound C & p`2 <= N-bound C & p`2 >= S-bound C
or p`1 <= E-bound C & p`1 >= W-bound C & p`2 = N-bound C or p`1 <= E-bound C &
p`1 >= W-bound C & p`2 = S-bound C or p`1 = E-bound C & p`2 <= N-bound C & p`2
  >= S-bound C);
  then
  x in LSeg(NW-corner C,NE-corner C) or x in LSeg(NE-corner C,SE-corner C
  ) or x in LSeg(SE-corner C,SW-corner C) or x in LSeg(SW-corner C,NW-corner C)
  by A1,A2,A4,A3;
  then x in (LSeg(NW-corner C,NE-corner C) \/ LSeg(NE-corner C,SE-corner C))
  or x in (LSeg(SE-corner C,SW-corner C) \/ LSeg(SW-corner C,NW-corner C)) by
XBOOLE_0:def 3;
  then x in (LSeg(NW-corner C,NE-corner C) \/ LSeg(NE-corner C,SE-corner C))
  \/ (LSeg(SE-corner C,SW-corner C) \/ LSeg(SW-corner C,NW-corner C)) by
XBOOLE_0:def 3;
  hence thesis by A5,SPRECT_1:41;
end;
