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 A1: r <= x by XXREAL_1:1;
assume y in [.r,s.] ; :: thesis: [.x,y.] c= [.r,s.]
then A2: y <= s by XXREAL_1:1;
let z be ExtReal; :: according to MEMBERED:def 8 :: thesis: ( not z in [.x,y.] or z in [.r,s.] )
assume A3: z in [.x,y.] ; :: thesis: z in [.r,s.]
then x <= z by XXREAL_1:1;
then A4: r <= z by A1, XXREAL_0:2;
z <= y by A3, XXREAL_1:1;
then z <= s by A2, XXREAL_0:2;
hence z in [.r,s.] by A4, XXREAL_1:1; :: thesis: verum