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

now :: thesis: not r in ].p,q.]

assume r in ].p,q.] ; :: thesis: contradiction
then A3: p < r by Th2;
s <= q by A1, A2, Th58;
then r <= q by A1, XXREAL_0:2;
then p < q by A3, XXREAL_0:2;
hence contradiction by A2, A3, Th53; :: thesis: verum

end;

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