let p, q, r, s be ExtReal; :: thesis: ( r <= p & s <= q implies ].r,s.] /\ ].p,q.] = ].p,s.] )
assume that
A1: r <= p and
A2: s <= q ; :: thesis: ].r,s.] /\ ].p,q.] = ].p,s.]
let t be ExtReal; :: according to MEMBERED:def 14 :: thesis: ( ( not t in ].r,s.] /\ ].p,q.] or t in ].p,s.] ) & ( not t in ].p,s.] or t in ].r,s.] /\ ].p,q.] ) )
thus ( t in ].r,s.] /\ ].p,q.] implies t in ].p,s.] ) :: thesis: ( not t in ].p,s.] or t in ].r,s.] /\ ].p,q.] )

proof

assume A3: t in ].r,s.] /\ ].p,q.] ; :: thesis: t in ].p,s.]
then A4: t in ].r,s.] by XBOOLE_0:def 4;
A5: t in ].p,q.] by A3, XBOOLE_0:def 4;
A6: t <= s by A4, Th2;
p < t by A5, Th2;
hence t in ].p,s.] by A6, Th2; :: thesis: verum

end;

assume A7: t in ].p,s.] ; :: thesis: t in ].r,s.] /\ ].p,q.]
then A8: p < t by Th2;
A9: t <= s by A7, Th2;
A10: r < t by A1, A8, XXREAL_0:2;
A11: t <= q by A2, A9, XXREAL_0:2;
A12: t in ].r,s.] by A9, A10, Th2;
t in ].p,q.] by A8, A11, Th2;
hence t in ].r,s.] /\ ].p,q.] by A12, XBOOLE_0:def 4; :: thesis: verum