let p be ext-real number ; :: thesis: [.+infty,p.[ = {}
for x being set holds not x in [.+infty,p.[

proof

given x being set such that A1: x in [.+infty,p.[ ; :: thesis: contradiction
reconsider s = x as ext-real number by A1;
A2: +infty <= s by A1, Th3;
s < p by A1, Th3;
then p > +infty by A2, XXREAL_0:2;
hence contradiction by XXREAL_0:3; :: thesis: verum

end;

hence [.+infty,p.[ = {} by XBOOLE_0:def 1; :: thesis: verum