let p, q, r, s be ExtReal; :: thesis: ( r < s & p <= q implies ].r,q.] \ ].p,s.[ = ].r,p.] \/ [.s,q.] )
assume that
A1: r < s and
A2: p <= q ; :: thesis: ].r,q.] \ ].p,s.[ = ].r,p.] \/ [.s,q.]
let x be ExtReal; :: 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 A3: x in ].r,q.] \ ].p,s.[ ; :: thesis: x in ].r,p.] \/ [.s,q.]
then A4: not x in ].p,s.[ by XBOOLE_0:def 5;
A5: r < x by A3, Th2;
A6: x <= q by A3, Th2;
( not p < x or not x < s ) by A4, Th4;
then ( x in ].r,p.] or x in [.s,q.] ) by A5, A6, Th1, Th2;
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 A7: ( ( r < x & x <= p ) or ( s <= x & x <= q ) ) by Th1, Th2;
then A8: r < x by A1, XXREAL_0:2;
x <= q by A2, A7, XXREAL_0:2;
then A9: x in ].r,q.] by A8, Th2;
not x in ].p,s.[ by A7, Th4;
hence x in ].r,q.] \ ].p,s.[ by A9, XBOOLE_0:def 5; :: thesis: verum