reserve m,k,j,j1,i,i1,i2,n for Nat,
  r,s for Real,
  C for compact non vertical non horizontal Subset of TOP-REAL 2,
  G for Go-board,
  p for Point of TOP-REAL 2;

theorem Th2:
  for p,q being Point of TOP-REAL 2 st p <> q holds Cl(LSeg(p,q) \
  {p,q}) = LSeg(p,q)
proof
  let p,q be Point of TOP-REAL 2 such that
A1: p <> q;
  Cl(LSeg(p,q) \ {p,q}) c= Cl LSeg(p,q) by PRE_TOPC:19,XBOOLE_1:36;
  hence Cl(LSeg(p,q) \ {p,q}) c= LSeg(p,q) by PRE_TOPC:22;
  let e be object;
  p in LSeg(p,q) & q in LSeg(p,q) by RLTOPSP1:68;
  then {p,q} c= LSeg(p,q) by ZFMISC_1:32;
  then
A2: LSeg(p,q) = LSeg(p,q) \ {p,q} \/ {p,q} by XBOOLE_1:45;
  assume e in LSeg(p,q);
  then
A3: e in LSeg(p,q) \ {p,q} or e in {p,q} by A2,XBOOLE_0:def 3;
  per cases by A3,TARSKI:def 2;
  suppose
A4: e in LSeg(p,q) \ {p,q};
    LSeg(p,q) \ {p,q} c= Cl(LSeg(p,q) \ {p,q}) by PRE_TOPC:18;
    hence thesis by A4;
  end;
  suppose
    e = p or e = q;
    hence thesis by A1,Th1;
  end;
end;
