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