let p, q, r, s be ExtReal; :: thesis: ].r,s.] /\ ].p,q.] = ].(max (r,p)),(min (s,q)).]
let t be ExtReal; :: according to MEMBERED:def 14 :: thesis: ( ( 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)).] ) :: thesis: ( 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.] ; :: thesis: 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, Th2;
A5: t <= s by A2, Th2;
A6: p < t by A3, Th2;
A7: t <= q by A3, Th2;
A8: max (r,p) < t by A4, A6, XXREAL_0:29;
t <= min (s,q) by A5, A7, XXREAL_0:20;
hence t in ].(max (r,p)),(min (s,q)).] by A8, Th2; :: thesis: verum

end;

assume A9: t in ].(max (r,p)),(min (s,q)).] ; :: thesis: t in ].r,s.] /\ ].p,q.]
then A10: max (r,p) < t by Th2;
A11: t <= min (s,q) by A9, Th2;
A12: r < t by A10, XXREAL_0:31;
A13: p < t by A10, XXREAL_0:31;
A14: t <= s by A11, XXREAL_0:22;
A15: t <= q by A11, XXREAL_0:22;
A16: t in ].r,s.] by A12, A14, Th2;
t in ].p,q.] by A13, A15, Th2;
hence t in ].r,s.] /\ ].p,q.] by A16, XBOOLE_0:def 4; :: thesis: verum