let C be non empty compact Subset of (TOP-REAL 2); :: thesis: (LSeg (NW-corner C),(SW-corner C)) /\ (LSeg (SW-corner C),(SE-corner C)) = {(SW-corner C)}
for a being set holds
( a in (LSeg (NW-corner C),(SW-corner C)) /\ (LSeg (SW-corner C),(SE-corner C)) iff a = SW-corner C )

proof

let a be set ; :: thesis: ( a in (LSeg (NW-corner C),(SW-corner C)) /\ (LSeg (SW-corner C),(SE-corner C)) iff a = SW-corner C )
thus ( a in (LSeg (NW-corner C),(SW-corner C)) /\ (LSeg (SW-corner C),(SE-corner C)) implies a = SW-corner C ) :: thesis: ( a = SW-corner C implies a in (LSeg (NW-corner C),(SW-corner C)) /\ (LSeg (SW-corner C),(SE-corner C)) )

proof

assume A1: a in (LSeg (NW-corner C),(SW-corner C)) /\ (LSeg (SW-corner C),(SE-corner C)) ; :: thesis: a = SW-corner C
then a in LSeg (NW-corner C),(SW-corner C) by XBOOLE_0:def 4;
then a in { p where p is Point of (TOP-REAL 2) : ( p `1 = W-bound C & p `2 <= N-bound C & p `2 >= S-bound C ) } by Th28;
then A2: ex p being Point of (TOP-REAL 2) st
( p = a & p `1 = W-bound C & p `2 <= N-bound C & p `2 >= S-bound C ) ;
a in LSeg (SW-corner C),(SE-corner C) by A1, XBOOLE_0:def 4;
then a in { p where p is Point of (TOP-REAL 2) : ( p `1 <= E-bound C & p `1 >= W-bound C & p `2 = S-bound C ) } by Th26;
then ex p being Point of (TOP-REAL 2) st
( p = a & p `1 <= E-bound C & p `1 >= W-bound C & p `2 = S-bound C ) ;
hence a = SW-corner C by A2, EUCLID:57; :: thesis: verum

end;

assume a = SW-corner C ; :: thesis: a in (LSeg (NW-corner C),(SW-corner C)) /\ (LSeg (SW-corner C),(SE-corner C))
then ( a in LSeg (NW-corner C),(SW-corner C) & a in LSeg (SW-corner C),(SE-corner C) ) by RLTOPSP1:69;
hence a in (LSeg (NW-corner C),(SW-corner C)) /\ (LSeg (SW-corner C),(SE-corner C)) by XBOOLE_0:def 4; :: thesis: verum

end;

hence (LSeg (NW-corner C),(SW-corner C)) /\ (LSeg (SW-corner C),(SE-corner C)) = {(SW-corner C)} by TARSKI:def 1; :: thesis: verum