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

proof

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

proof

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

end;

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

end;

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