let p, q, r, s be ExtReal; [.r,s.] /\ [.p,q.] = [.(max (r,p)),(min (s,q)).]
let t be ExtReal; MEMBERED:def 14 ( ( not t in [.r,s.] /\ [.p,q.] or t in [.(max (r,p)),(min (s,q)).] ) & ( not t in [.(max (r,p)),(min (s,q)).] or t in [.r,s.] /\ [.p,q.] ) )
thus
( t in [.r,s.] /\ [.p,q.] implies t in [.(max (r,p)),(min (s,q)).] )
( not t in [.(max (r,p)),(min (s,q)).] or t in [.r,s.] /\ [.p,q.] )proof
assume A1:
t in [.r,s.] /\ [.p,q.]
;
t in [.(max (r,p)),(min (s,q)).]
then A2:
t in [.r,s.]
by XBOOLE_0:def 4;
A3:
t in [.p,q.]
by A1, XBOOLE_0:def 4;
A4:
r <= t
by A2, Th1;
A5:
t <= s
by A2, Th1;
A6:
p <= t
by A3, Th1;
A7:
t <= q
by A3, Th1;
A8:
max (
r,
p)
<= t
by A4, A6, XXREAL_0:28;
t <= min (
s,
q)
by A5, A7, XXREAL_0:20;
hence
t in [.(max (r,p)),(min (s,q)).]
by A8, Th1;
verum
end;
assume A9:
t in [.(max (r,p)),(min (s,q)).]
; t in [.r,s.] /\ [.p,q.]
then A10:
max (r,p) <= t
by Th1;
A11:
t <= min (s,q)
by A9, Th1;
A12:
r <= t
by A10, XXREAL_0:30;
A13:
p <= t
by A10, XXREAL_0:30;
A14:
t <= s
by A11, XXREAL_0:22;
A15:
t <= q
by A11, XXREAL_0:22;
A16:
t in [.r,s.]
by A12, A14, Th1;
t in [.p,q.]
by A13, A15, Th1;
hence
t in [.r,s.] /\ [.p,q.]
by A16, XBOOLE_0:def 4; verum