let p, q, r, s be ext-real number ; :: 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 ext-real number such that
A1: u in [.r,s.[ and
A2: u in [.p,q.[ by MEMBERED:def 20;
A3: ( r <= u & u <= s ) by A1, Th3;
A4: ( p <= u & u <= q ) by A2, Th3;
then A5: r <= q by A3, XXREAL_0:2;
A6: p <= s by A3, A4, XXREAL_0:2;
let t be ext-real number ; :: 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 t in [.p,q.[ \ [.r,s.[ ; :: thesis: t in [.p,r.[ \/ [.s,q.[
then ( t in [.p,q.[ & not t in [.r,s.[ ) by XBOOLE_0:def 5;
then ( p <= t & t < q & ( t < r or s <= t ) ) by Th3;
then ( t in [.p,r.[ or t in [.s,q.[ ) by 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 ( ( p <= t & t < r ) or ( s <= t & t < q ) ) by Th3;
then ( p <= t & t < q & ( t < r or s <= t ) ) by A5, A6, XXREAL_0:2;
then ( t in [.p,q.[ & not t in [.r,s.[ ) by Th3;
hence t in [.p,q.[ \ [.r,s.[ by XBOOLE_0:def 5; :: thesis: verum