reserve T,T1,T2,S for non empty TopSpace;
reserve p,q for Point of TOP-REAL 2;

theorem Th13:
  for p being Point of TOP-REAL 2 st not(p`2<=p`1 & -p`1<=p`2 or p
  `2>=p`1 & p`2<=-p`1) holds p`1<=p`2 & -p`2<=p`1 or p`1>=p`2 & p`1<=-p`2
proof
  let p being Point of TOP-REAL 2;
A1: -p`1<p`2 implies --p`1>-p`2 by XREAL_1:24;
A2: -p`1>p`2 implies --p`1<-p`2 by XREAL_1:24;
  assume not(p`2<=p`1 & -p`1<=p`2 or p`2>=p`1 & p`2<=-p`1);
  hence thesis by A1,A2;
end;
