let r, s, p, q be ext-real number ; :: thesis: ( r <= s implies [.r,s.] <> ].p,q.[ )
assume that
A1: r <= s and
A2: [.r,s.] = ].p,q.[ ; :: thesis: contradiction

now

assume s in ].p,q.[ ; :: thesis: contradiction
then A3: s < q by Th4;
p <= r by A1, A2, Th62;
then p <= s by A1, XXREAL_0:2;
then p < q by A3, XXREAL_0:2;
hence contradiction by A2, A3, Th51; :: thesis: verum

end;

hence contradiction by A1, A2, Th1; :: thesis: verum