let p, q, r be Point of (TOP-REAL 2); :: thesis: ( LSeg p,q is vertical & LSeg q,r is horizontal implies (LSeg p,q) /\ (LSeg q,r) = {q} )
assume that
A1: LSeg p,q is vertical and
A2: LSeg q,r is horizontal ; :: thesis: (LSeg p,q) /\ (LSeg q,r) = {q}
for x being set holds
( x in (LSeg p,q) /\ (LSeg q,r) iff x = q )

proof

let x be set ; :: thesis: ( x in (LSeg p,q) /\ (LSeg q,r) iff x = q )
thus ( x in (LSeg p,q) /\ (LSeg q,r) implies x = q ) :: thesis: ( x = q implies x in (LSeg p,q) /\ (LSeg q,r) )

proof

assume A3: x in (LSeg p,q) /\ (LSeg q,r) ; :: thesis: x = q
then reconsider x = x as Point of (TOP-REAL 2) ;
x in LSeg q,r by A3, XBOOLE_0:def 4;
then A4: x `2 = q `2 by A2, SPPOL_1:63;
x in LSeg p,q by A3, XBOOLE_0:def 4;
then x `1 = q `1 by A1, SPPOL_1:64;
hence x = q by A4, TOPREAL3:11; :: thesis: verum

end;

assume A5: x = q ; :: thesis: x in (LSeg p,q) /\ (LSeg q,r)
then A6: x in LSeg q,r by RLTOPSP1:69;
x in LSeg p,q by A5, RLTOPSP1:69;
hence x in (LSeg p,q) /\ (LSeg q,r) by A6, XBOOLE_0:def 4; :: thesis: verum

end;

hence (LSeg p,q) /\ (LSeg q,r) = {q} by TARSKI:def 1; :: thesis: verum