let p, q, r, s be ExtReal; ( r <= p & s <= q implies ].r,s.] /\ ].p,q.] = ].p,s.] )
assume that
A1:
r <= p
and
A2:
s <= q
; ].r,s.] /\ ].p,q.] = ].p,s.]
let t be ExtReal; MEMBERED:def 14 ( ( 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.] )
( not t in ].p,s.] or t in ].r,s.] /\ ].p,q.] )proof
assume A3:
t in ].r,s.] /\ ].p,q.]
;
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;
verum
end;
assume A7:
t in ].p,s.]
; 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; verum