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 being rectangular FinSequence of TOP-REAL 2, p,q being Point of
TOP-REAL 2 st p in L~f & not q in L~f & <*q*> is_in_the_area_of f holds LSeg(p,
  q) /\ L~f = {p}
proof
  let f be rectangular FinSequence of TOP-REAL 2, p,q be Point of TOP-REAL 2
  such that
A1: p in L~f and
A2: not q in L~f and
A3: <*q*> is_in_the_area_of f;
A4: <*p,q*> = <*p*>^<*q*> by FINSEQ_1:def 9;
  <*p*> is_in_the_area_of f by A1,Th46,SPRECT_2:21;
  hence LSeg(p,q) /\ L~f c= {p} by A2,A3,A4,Th47,SPRECT_2:24;
  p in LSeg(p,q) by RLTOPSP1:68;
  then p in LSeg(p,q) /\ L~f by A1,XBOOLE_0:def 4;
  hence thesis by ZFMISC_1:31;
end;
