consider x being Element of (LSeg |[0 ,0 ]|,|[1,0 ]|) /\ (LSeg |[0 ,1]|,|[1,1]|);
assume A1: (LSeg |[0 ,0 ]|,|[1,0 ]|) /\ (LSeg |[0 ,1]|,|[1,1]|) <> {} ; :: according to XBOOLE_0:def 7 :: thesis: contradiction
then x in LSeg |[0 ,1]|,|[1,1]| by XBOOLE_0:def 4;
then A2: ex p being Point of (TOP-REAL 2) st
( p = x & p `1 <= 1 & p `1 >= 0 & p `2 = 1 ) by Th19;
x in LSeg |[0 ,0 ]|,|[1,0 ]| by A1, XBOOLE_0:def 4;
then ex p2 being Point of (TOP-REAL 2) st
( p2 = x & p2 `1 <= 1 & p2 `1 >= 0 & p2 `2 = 0 ) by Th19;
hence contradiction by A2; :: thesis: verum