let s, p, r, q be ext-real number ; :: thesis: ( s <= p implies ].r,s.[ misses ].p,q.[ )
assume A1: s <= p ; :: thesis: ].r,s.[ misses ].p,q.[
let t be real number ; :: according to MEMBERED:def 21 :: thesis: ( not t in ].r,s.[ or not t in ].p,q.[ )
assume t in ].r,s.[ ; :: thesis: not t in ].p,q.[
then t <= s by Th4;
then t <= p by A1, XXREAL_0:2;
hence not t in ].p,q.[ by Th4; :: thesis: verum