let p, r, s, q be ext-real number ; :: thesis: ( p <= r & s < q implies ].r,s.] c= ].p,q.[ )
assume A1: ( p <= r & s < q ) ; :: thesis: ].r,s.] c= ].p,q.[
let t be ext-real number ; :: according to MEMBERED:def 8 :: thesis: ( not t in ].r,s.] or t in ].p,q.[ )
assume t in ].r,s.] ; :: thesis: t in ].p,q.[
then ( r < t & t <= s ) by Th2;
then ( p < t & t < q ) by A1, XXREAL_0:2;
hence t in ].p,q.[ by Th4; :: thesis: verum