let r, s, t be ext-real number ; :: thesis: ( r <= s implies [.r,t.] \ [.r,s.] = ].s,t.] )
assume A1: r <= s ; :: thesis: [.r,t.] \ [.r,s.] = ].s,t.]
let p be ext-real number ; :: according to MEMBERED:def 14 :: thesis: ( ( not p in [.r,t.] \ [.r,s.] or p in ].s,t.] ) & ( not p in ].s,t.] or p in [.r,t.] \ [.r,s.] ) )
thus ( p in [.r,t.] \ [.r,s.] implies p in ].s,t.] ) :: thesis: ( not p in ].s,t.] or p in [.r,t.] \ [.r,s.] )

proof

assume p in [.r,t.] \ [.r,s.] ; :: thesis: p in ].s,t.]
then ( p in [.r,t.] & not p in [.r,s.] ) by XBOOLE_0:def 5;
then ( r <= p & p <= t & ( p < r or s < p ) ) by Th1;
hence p in ].s,t.] by Th2; :: thesis: verum

end;

assume p in ].s,t.] ; :: thesis: p in [.r,t.] \ [.r,s.]
then ( s < p & p <= t ) by Th2;
then ( r <= p & p <= t & ( p < r or s < p ) ) by A1, XXREAL_0:2;
then ( p in [.r,t.] & not p in [.r,s.] ) by Th1;
hence p in [.r,t.] \ [.r,s.] by XBOOLE_0:def 5; :: thesis: verum