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 Th4;
then ( p <= t & t < q ) by A1, XXREAL_0:2;
hence t in [.p,q.[ by Th3; :: thesis: verum