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 Th2, Th3;
then ( p < t & t < q ) by A1, A2, XXREAL_0:2;
hence t in ].p,q.[ by Th4; :: 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, Th4, XXREAL_0:2;
then ( t in ].p,r.] or t in [.s,q.[ ) by Th2, Th3;
hence t in ].p,r.] \/ [.s,q.[ by XBOOLE_0:def 3; :: thesis: verum