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, Th4;
A5:
t < s
by A2, Th4;
A6:
p < t
by A3, Th4;
A7:
t < q
by A3, Th4;
A8:
max (
r,
p)
< t
by A4, A6, XXREAL_0:29;
t < min (
s,
q)
by A5, A7, XXREAL_0:21;
hence
t in ].(max (r,p)),(min (s,q)).[
by A8, Th4;
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 Th4;
A11:
t < min (s,q)
by A9, Th4;
A12:
r < t
by A10, XXREAL_0:31;
A13:
p < t
by A10, XXREAL_0:31;
A14:
t < s
by A11, XXREAL_0:23;
A15:
t < q
by A11, XXREAL_0:23;
A16:
t in ].r,s.[
by A12, A14, Th4;
t in ].p,q.[
by A13, A15, Th4;
hence
t in ].r,s.[ /\ ].p,q.[
by A16, XBOOLE_0:def 4; verum