let x be ExtReal; :: according to XXREAL_2:def 12 :: thesis: for s being ExtReal st x in ].r,s.] & s in ].r,s.] holds
[.x,s.] c= ].r,s.]
let y be ExtReal; :: thesis: ( x in ].r,s.] & y in ].r,s.] implies [.x,y.] c= ].r,s.] )
assume x in ].r,s.] ; :: thesis: ( not y in ].r,s.] or [.x,y.] c= ].r,s.] )
then A5: r < x by XXREAL_1:2;
assume y in ].r,s.] ; :: thesis: [.x,y.] c= ].r,s.]
then A6: y <= s by XXREAL_1:2;
let z be ExtReal; :: according to MEMBERED:def 8 :: thesis: ( not z in [.x,y.] or z in ].r,s.] )
assume A7: z in [.x,y.] ; :: thesis: z in ].r,s.]
then x <= z by XXREAL_1:1;
then A8: r < z by A5, XXREAL_0:2;
z <= y by A7, XXREAL_1:1;
then z <= s by A6, XXREAL_0:2;
hence z in ].r,s.] by A8, XXREAL_1:2; :: thesis: verum