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, 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; :: 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 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; :: thesis: verum