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 Th43:
  for f being rectangular FinSequence of TOP-REAL 2, p being Point
  of TOP-REAL 2 st <*p*> is_in_the_area_of f & (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) holds p in L~f
proof
  let f be rectangular FinSequence of TOP-REAL 2, p be Point of TOP-REAL 2;
A1: <*p*>/.1 = p by FINSEQ_4:16;
  dom<*p*> = {1} by FINSEQ_1:2,38;
  then
A2: 1 in dom<*p*> by TARSKI:def 1;
  assume
A3: <*p*> is_in_the_area_of f;
  then
A4: W-bound L~f <= p`1 by A2,A1;
A5: p`2 <= N-bound L~f by A3,A2,A1;
A6: S-bound L~f <= p`2 by A3,A2,A1;
A7: p`1 <= E-bound L~f by A3,A2,A1;
  consider D being non vertical non horizontal non empty compact Subset of
  TOP-REAL 2 such that
A8: f = SpStSeq D by SPRECT_1:def 2;
A9: E-bound L~SpStSeq D = E-bound D by SPRECT_1:61;
A10: N-bound L~SpStSeq D = N-bound D by SPRECT_1:60;
A11: S-bound L~SpStSeq D = S-bound D by SPRECT_1:59;
A12: W-bound L~SpStSeq D = W-bound D by SPRECT_1:58;
A13: 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 A8,
SPRECT_1:41;
  assume
A14: 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;
  per cases by A14;
  suppose
A15: p`1 = W-bound L~f;
A16: (NW-corner D)`1 = W-bound D by EUCLID:52;
A17: (NW-corner D)`2 = N-bound D by EUCLID:52;
A18: (SW-corner D)`2 = S-bound D by EUCLID:52;
    (SW-corner D)`1 = W-bound D by EUCLID:52;
    then p in LSeg(SW-corner D,NW-corner D) by A6,A5,A8,A12,A11,A10,A15,A16,A18
,A17,GOBOARD7:7;
    then p in LSeg(SE-corner D,SW-corner D) \/ LSeg(SW-corner D,NW-corner D)
    by XBOOLE_0:def 3;
    hence thesis by A13,XBOOLE_0:def 3;
  end;
  suppose
A19: p`1 = E-bound L~f;
A20: (SE-corner D)`1 = E-bound D by EUCLID:52;
A21: (SE-corner D)`2 = S-bound D by EUCLID:52;
A22: (NE-corner D)`2 = N-bound D by EUCLID:52;
    (NE-corner D)`1 = E-bound D by EUCLID:52;
    then p in LSeg(NE-corner D,SE-corner D) by A6,A5,A8,A11,A10,A9,A19,A20,A22
,A21,GOBOARD7:7;
    then p in LSeg(NW-corner D,NE-corner D) \/ LSeg(NE-corner D,SE-corner D)
    by XBOOLE_0:def 3;
    hence thesis by A13,XBOOLE_0:def 3;
  end;
  suppose
A23: p`2 = S-bound L~f;
A24: (SW-corner D)`1 = W-bound D by EUCLID:52;
A25: (SW-corner D)`2 = S-bound D by EUCLID:52;
A26: (SE-corner D)`2 = S-bound D by EUCLID:52;
    (SE-corner D)`1 = E-bound D by EUCLID:52;
    then p in LSeg(SE-corner D,SW-corner D) by A4,A7,A8,A12,A11,A9,A23,A24,A26
,A25,GOBOARD7:8;
    then p in LSeg(SE-corner D,SW-corner D) \/ LSeg(SW-corner D,NW-corner D)
    by XBOOLE_0:def 3;
    hence thesis by A13,XBOOLE_0:def 3;
  end;
  suppose
A27: p`2 = N-bound L~f;
A28: (NE-corner D)`1 = E-bound D by EUCLID:52;
A29: (NE-corner D)`2 = N-bound D by EUCLID:52;
A30: (NW-corner D)`2 = N-bound D by EUCLID:52;
    (NW-corner D)`1 = W-bound D by EUCLID:52;
    then p in LSeg(NW-corner D,NE-corner D) by A4,A7,A8,A12,A10,A9,A27,A28,A30
,A29,GOBOARD7:8;
    then p in LSeg(NW-corner D,NE-corner D) \/ LSeg(NE-corner D,SE-corner D)
    by XBOOLE_0:def 3;
    hence thesis by A13,XBOOLE_0:def 3;
  end;
end;
