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 real number ; :: according to MEMBERED:def 15 :: 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, Th1;
then ( x in ].r,p.[ or x in ].s,q.[ ) by Th4;
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 Th4;
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, Th1;
hence x in ].r,q.[ \ [.p,s.] by XBOOLE_0:def 5; :: thesis: verum