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 Th3;
then t <= p by A1, XXREAL_0:2;
hence not t in ].p,q.] by Th2; :: thesis: verum