let r, s, p, q be ext-real number ; :: thesis: [.r,s.] /\ [.p,q.] = [.(max r,p),(min s,q).]
let t be ext-real number ; :: 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, 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; :: 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 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; :: thesis: verum