let x be ext-real number ; :: according to XXREAL_2:def 12 :: thesis: for s being ext-real number st x in ].r,s.[ & s in ].r,s.[ holds
[.x,s.] c= ].r,s.[
let y be ext-real number ; :: 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 A14: r < x by XXREAL_1:4;
assume y in ].r,s.[ ; :: thesis: [.x,y.] c= ].r,s.[
then A15: y < s by XXREAL_1:4;
let z be ext-real number ; :: according to MEMBERED:def 8 :: thesis: ( not z in [.x,y.] or z in ].r,s.[ )
assume A16: z in [.x,y.] ; :: thesis: z in ].r,s.[
then x <= z by XXREAL_1:1;
then A17: r < z by A14, XXREAL_0:2;
z <= y by A16, XXREAL_1:1;
then z < s by A15, XXREAL_0:2;
hence z in ].r,s.[ by A17, XXREAL_1:4; :: thesis: verum