let p, q, r, s be ExtReal; :: thesis: ( p <= s & s <= r & r < q implies [.p,r.] \/ [.s,q.[ = [.p,q.[ )
assume that
A1: p <= s and
A2: s <= r and
A3: r < q ; :: thesis: [.p,r.] \/ [.s,q.[ = [.p,q.[
let t be ExtReal; :: 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 A4: ( ( p <= t & t <= r ) or ( s <= t & t < q ) ) by Th1, Th3;
then A5: p <= t by A1, XXREAL_0:2;
t < q by A3, A4, XXREAL_0:2;
hence t in [.p,q.[ by A5, Th3; :: 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 A2, Th3, XXREAL_0:2;
then ( t in [.p,r.] or t in [.s,q.[ ) by Th1, Th3;
hence t in [.p,r.] \/ [.s,q.[ by XBOOLE_0:def 3; :: thesis: verum