let r, s be ExtReal; :: thesis: ( r < s implies ].r,s.] = ].r,s.[ \/ {s} )
assume A1: r < s ; :: thesis: ].r,s.] = ].r,s.[ \/ {s}
let t be ExtReal; :: according to MEMBERED:def 14 :: thesis: ( ( not t in ].r,s.] or t in ].r,s.[ \/ {s} ) & ( not t in ].r,s.[ \/ {s} or t in ].r,s.] ) )
thus ( t in ].r,s.] implies t in ].r,s.[ \/ {s} ) :: thesis: ( not t in ].r,s.[ \/ {s} or t in ].r,s.] )

proof

assume t in ].r,s.] ; :: thesis: t in ].r,s.[ \/ {s}
then ( t in ].r,s.[ or t = s ) by Th9;
then ( t in ].r,s.[ or t in {s} ) by TARSKI:def 1;
hence t in ].r,s.[ \/ {s} by XBOOLE_0:def 3; :: thesis: verum

end;

assume t in ].r,s.[ \/ {s} ; :: thesis: t in ].r,s.]
then ( t in ].r,s.[ or t in {s} ) by XBOOLE_0:def 3;
then ( t in ].r,s.[ or t = s ) by TARSKI:def 1;
hence t in ].r,s.] by A1, Th2, Th15; :: thesis: verum