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 ext-real number ; :: according to MEMBERED:def 20 :: 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 Th1;
then t < p by A1, XXREAL_0:2;
hence not t in [.p,q.[ by Th3; :: thesis: verum