let p, q, r, s be ExtReal; :: thesis: ( [.p,q.[ meets [.r,s.[ implies [.p,q.[ \ [.r,s.[ = [.p,r.[ \/ [.s,q.[ )
assume [.p,q.[ meets [.r,s.[ ; :: thesis: [.p,q.[ \ [.r,s.[ = [.p,r.[ \/ [.s,q.[
then consider u being ExtReal such that
A1: u in [.r,s.[ and
A2: u in [.p,q.[ ;
A3: r <= u by A1, Th3;
A4: u <= s by A1, Th3;
A5: p <= u by A2, Th3;
u <= q by A2, Th3;
then A6: r <= q by A3, XXREAL_0:2;
A7: p <= s by A4, A5, XXREAL_0:2;
let t be ExtReal; :: according to MEMBERED:def 14 :: thesis: ( ( not t in [.p,q.[ \ [.r,s.[ or t in [.p,r.[ \/ [.s,q.[ ) & ( not t in [.p,r.[ \/ [.s,q.[ or t in [.p,q.[ \ [.r,s.[ ) )
thus ( t in [.p,q.[ \ [.r,s.[ implies t in [.p,r.[ \/ [.s,q.[ ) :: thesis: ( not t in [.p,r.[ \/ [.s,q.[ or t in [.p,q.[ \ [.r,s.[ )

proof

assume A8: t in [.p,q.[ \ [.r,s.[ ; :: thesis: t in [.p,r.[ \/ [.s,q.[
then A9: not t in [.r,s.[ by XBOOLE_0:def 5;
A10: p <= t by A8, Th3;
A11: t < q by A8, Th3;
( t < r or s <= t ) by A9, Th3;
then ( t in [.p,r.[ or t in [.s,q.[ ) by A10, A11, Th3;
hence t in [.p,r.[ \/ [.s,q.[ by XBOOLE_0:def 3; :: thesis: verum

end;

assume t in [.p,r.[ \/ [.s,q.[ ; :: thesis: t in [.p,q.[ \ [.r,s.[
then ( t in [.p,r.[ or t in [.s,q.[ ) by XBOOLE_0:def 3;
then A12: ( ( p <= t & t < r ) or ( s <= t & t < q ) ) by Th3;
then A13: p <= t by A7, XXREAL_0:2;
t < q by A6, A12, XXREAL_0:2;
then A14: t in [.p,q.[ by A13, Th3;
not t in [.r,s.[ by A12, Th3;
hence t in [.p,q.[ \ [.r,s.[ by A14, XBOOLE_0:def 5; :: thesis: verum