let p, s, r, q be ext-real number ; :: thesis: ( p < s & r <= q & s <= r implies ].p,r.] \/ [.s,q.] = ].p,q.] )
assume that
A1: p < s and
A2: r <= q and
A3: s <= r ; :: thesis: ].p,r.] \/ [.s,q.] = ].p,q.]
let t be ext-real number ; :: according to MEMBERED:def 14 :: thesis: ( ( not t in ].p,r.] \/ [.s,q.] or t in ].p,q.] ) & ( not t in ].p,q.] or t in ].p,r.] \/ [.s,q.] ) )
thus ( t in ].p,r.] \/ [.s,q.] implies t in ].p,q.] ) :: thesis: ( not t in ].p,q.] or t in ].p,r.] \/ [.s,q.] )

proof

assume t in ].p,r.] \/ [.s,q.] ; :: thesis: t in ].p,q.]
then ( t in ].p,r.] or t in [.s,q.] ) by XBOOLE_0:def 3;
then ( ( p < t & t <= r ) or ( s <= t & t <= q ) ) by Th1, Th2;
then ( p < t & t <= q ) by A1, A2, XXREAL_0:2;
hence t in ].p,q.] by Th2; :: thesis: verum

end;

assume t in ].p,q.] ; :: thesis: t in ].p,r.] \/ [.s,q.]
then ( ( p < t & t <= r ) or ( s <= t & t <= q ) ) by A3, Th2, XXREAL_0:2;
then ( t in ].p,r.] or t in [.s,q.] ) by Th1, Th2;
hence t in ].p,r.] \/ [.s,q.] by XBOOLE_0:def 3; :: thesis: verum