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 Th1;
then ( p <= t & t < q ) by A1, XXREAL_0:2;
hence t in [.p,q.[ by Th3; :: thesis: verum