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 Th32:
  for f being rectangular FinSequence of TOP-REAL 2, p being Point
of TOP-REAL 2 st p in L~f holds p`1 = W-bound L~f or p`1 = E-bound L~f or p`2 =
  S-bound L~f or p`2 = N-bound L~f
proof
  let f be rectangular FinSequence of TOP-REAL 2, p be Point of TOP-REAL 2
  such that
A1: p in L~f;
  consider D being non vertical non horizontal non empty compact Subset of
  TOP-REAL 2 such that
A2: f = SpStSeq D by SPRECT_1:def 2;
  L~f = (LSeg(NW-corner D,NE-corner D) \/ LSeg(NE-corner D,SE-corner D))
  \/ (LSeg(SE-corner D,SW-corner D) \/ LSeg(SW-corner D,NW-corner D)) by A2,
SPRECT_1:41;
  then
A3: p in LSeg(NW-corner D,NE-corner D) \/ LSeg(NE-corner D,SE-corner D ) or
  p in LSeg(SE-corner D,SW-corner D) \/ LSeg(SW-corner D,NW-corner D) by A1,
XBOOLE_0:def 3;
  per cases by A3,XBOOLE_0:def 3;
  suppose
A4: p in LSeg(NW-corner D,NE-corner D);
A5: N-bound L~SpStSeq D = N-bound D by SPRECT_1:60;
    then
A6: (NE-corner D)`2 = N-bound L~f by A2,EUCLID:52;
    (NW-corner D)`2 = N-bound L~f by A2,A5,EUCLID:52;
    hence thesis by A4,A6,GOBOARD7:6;
  end;
  suppose
A7: p in LSeg(NE-corner D,SE-corner D);
A8: E-bound L~SpStSeq D = E-bound D by SPRECT_1:61;
    then
A9: (SE-corner D)`1 = E-bound L~f by A2,EUCLID:52;
    (NE-corner D)`1 = E-bound L~f by A2,A8,EUCLID:52;
    hence thesis by A7,A9,GOBOARD7:5;
  end;
  suppose
A10: p in LSeg(SE-corner D,SW-corner D);
A11: S-bound L~SpStSeq D = S-bound D by SPRECT_1:59;
    then
A12: (SW-corner D)`2 = S-bound L~f by A2,EUCLID:52;
    (SE-corner D)`2 = S-bound L~f by A2,A11,EUCLID:52;
    hence thesis by A10,A12,GOBOARD7:6;
  end;
  suppose
A13: p in LSeg(SW-corner D,NW-corner D);
A14: W-bound L~SpStSeq D = W-bound D by SPRECT_1:58;
    then
A15: (NW-corner D)`1 = W-bound L~f by A2,EUCLID:52;
    (SW-corner D)`1 = W-bound L~f by A2,A14,EUCLID:52;
    hence thesis by A13,A15,GOBOARD7:5;
  end;
end;
