let r, s, p, q be ext-real number ; :: thesis: ( r < s & p < q implies ].r,q.[ \ ].p,s.[ = ].r,p.] \/ [.s,q.[ )
assume that
Z1: r < s and
Z3: p < q ; :: thesis: ].r,q.[ \ ].p,s.[ = ].r,p.] \/ [.s,q.[
let x be ext-real number ; :: according to MEMBERED:def 14 :: thesis: ( ( not x in ].r,q.[ \ ].p,s.[ or x in ].r,p.] \/ [.s,q.[ ) & ( not x in ].r,p.] \/ [.s,q.[ or x in ].r,q.[ \ ].p,s.[ ) )
thus ( x in ].r,q.[ \ ].p,s.[ implies x in ].r,p.] \/ [.s,q.[ ) :: thesis: ( not x in ].r,p.] \/ [.s,q.[ or x in ].r,q.[ \ ].p,s.[ )

proof

assume x in ].r,q.[ \ ].p,s.[ ; :: thesis: x in ].r,p.] \/ [.s,q.[
then ( x in ].r,q.[ & not x in ].p,s.[ ) by XBOOLE_0:def 5;
then ( r < x & x < q & ( not p < x or not x < s ) ) by Th4;
then ( x in ].r,p.] or x in [.s,q.[ ) by Th2, Th3;
hence x in ].r,p.] \/ [.s,q.[ by XBOOLE_0:def 3; :: thesis: verum

end;

assume x in ].r,p.] \/ [.s,q.[ ; :: thesis: x in ].r,q.[ \ ].p,s.[
then ( x in ].r,p.] or x in [.s,q.[ ) by XBOOLE_0:def 3;
then ( ( r < x & x <= p ) or ( s <= x & x < q ) ) by Th2, Th3;
then ( r < x & x < q & ( x <= p or s <= x ) ) by Z1, Z3, XXREAL_0:2;
then ( x in ].r,q.[ & not x in ].p,s.[ ) by Th4;
hence x in ].r,q.[ \ ].p,s.[ by XBOOLE_0:def 5; :: thesis: verum