reserve S for Subset of TOP-REAL 2,
  C,C1,C2 for non empty compact Subset of TOP-REAL 2,
  p,q for Point of TOP-REAL 2;
reserve i,j,k for Nat,
  t,r1,r2,s1,s2 for Real;
reserve D1 for non vertical non empty compact Subset of TOP-REAL 2,
  D2 for non horizontal non empty compact Subset of TOP-REAL 2,
  D for non vertical non horizontal non empty compact Subset of TOP-REAL 2;

theorem Th34:
  LSeg(SW-corner D2,SE-corner D2) misses LSeg(NW-corner D2, NE-corner D2)
proof
  assume
  LSeg(SW-corner D2,SE-corner D2) /\ LSeg(NW-corner D2,NE-corner D2) <> {};
  then consider a being object such that
A1: a in LSeg(SW-corner D2,SE-corner D2) /\ LSeg(NW-corner D2,NE-corner
  D2) by XBOOLE_0:def 1;
  a in LSeg(NE-corner D2,NW-corner D2) by A1,XBOOLE_0:def 4;
  then
  a in {p : p`1 <= E-bound D2 & p`1 >= W-bound D2 & p`2= N-bound D2} by Th25;
  then
A2: ex p st p=a & p`1 <= E-bound D2 & p`1 >= W-bound D2 & p`2 = N-bound D2;
  a in LSeg(SE-corner D2,SW-corner D2)by A1,XBOOLE_0:def 4;
  then a in {p : p`1 <= E-bound D2 & p`1 >= W-bound D2 & p`2 = S-bound D2} by
Th24;
  then
  ex p st p = a & p`1 <= E-bound D2 & p`1 >= W-bound D2 & p`2 = S-bound D2;
  hence contradiction by A2,Th16;
end;
