for a being set holds
( a in (LSeg |[0 ,0 ]|,|[1,0 ]|) /\ (LSeg |[1,0 ]|,|[1,1]|) iff a = |[1,0 ]| )

proof

let a be set ; :: thesis: ( a in (LSeg |[0 ,0 ]|,|[1,0 ]|) /\ (LSeg |[1,0 ]|,|[1,1]|) iff a = |[1,0 ]| )
set p00 = |[0 ,0 ]|;
set p10 = |[1,0 ]|;
set p11 = |[1,1]|;
thus ( a in (LSeg |[0 ,0 ]|,|[1,0 ]|) /\ (LSeg |[1,0 ]|,|[1,1]|) implies a = |[1,0 ]| ) :: thesis: ( a = |[1,0 ]| implies a in (LSeg |[0 ,0 ]|,|[1,0 ]|) /\ (LSeg |[1,0 ]|,|[1,1]|) )

proof

assume A1: a in (LSeg |[0 ,0 ]|,|[1,0 ]|) /\ (LSeg |[1,0 ]|,|[1,1]|) ; :: thesis: a = |[1,0 ]|
then A2: a in LSeg |[1,0 ]|,|[1,1]| by XBOOLE_0:def 4;
a in { p where p is Point of (TOP-REAL 2) : ( p `1 <= 1 & p `1 >= 0 & p `2 = 0 ) } by A1, Th19, XBOOLE_0:def 4;
then A3: ex p being Point of (TOP-REAL 2) st
( p = a & p `1 <= 1 & p `1 >= 0 & p `2 = 0 ) ;
ex p2 being Point of (TOP-REAL 2) st
( p2 = a & p2 `1 = 1 & p2 `2 <= 1 & p2 `2 >= 0 ) by A2, Th19;
hence a = |[1,0 ]| by A3, EUCLID:57; :: thesis: verum

end;

assume a = |[1,0 ]| ; :: thesis: a in (LSeg |[0 ,0 ]|,|[1,0 ]|) /\ (LSeg |[1,0 ]|,|[1,1]|)
then ( a in LSeg |[0 ,0 ]|,|[1,0 ]| & a in LSeg |[1,0 ]|,|[1,1]| ) by RLTOPSP1:69;
hence a in (LSeg |[0 ,0 ]|,|[1,0 ]|) /\ (LSeg |[1,0 ]|,|[1,1]|) by XBOOLE_0:def 4; :: thesis: verum

end;

hence (LSeg |[0 ,0 ]|,|[1,0 ]|) /\ (LSeg |[1,0 ]|,|[1,1]|) = {|[1,0 ]|} by TARSKI:def 1; :: thesis: verum