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 A1: r <= x by XXREAL_1:3;
assume y in [.r,s.[ ; :: thesis: [.x,y.] c= [.r,s.[
then A2: y < s by XXREAL_1:3;
let z be ext-real number ; :: according to MEMBERED:def 8 :: thesis: ( not z in [.x,y.] or z in [.r,s.[ )
assume z in [.x,y.] ; :: thesis: z in [.r,s.[
then ( x <= z & z <= y ) by XXREAL_1:1;
then ( r <= z & z < s ) by A1, A2, XXREAL_0:2;
hence z in [.r,s.[ by XXREAL_1:3; :: thesis: verum