let r, s, t be ExtReal; :: thesis: ( r <= s implies [.r,t.] \ [.r,s.] = ].s,t.] )
assume A1: r <= s ; :: thesis: [.r,t.] \ [.r,s.] = ].s,t.]
let p be ExtReal; :: 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 A2: p in [.r,t.] \ [.r,s.] ; :: thesis: p in ].s,t.]
then A3: not p in [.r,s.] by XBOOLE_0:def 5;
A4: p <= t by A2, Th1;
( p < r or s < p ) by A3, Th1;
hence p in ].s,t.] by A2, A4, Th1, Th2; :: thesis: verum

end;

assume A5: p in ].s,t.] ; :: thesis: p in [.r,t.] \ [.r,s.]
then A6: s < p by Th2;
then A7: r <= p by A1, XXREAL_0:2;
p <= t by A5, Th2;
then A8: p in [.r,t.] by A7, Th1;
not p in [.r,s.] by A6, Th1;
hence p in [.r,t.] \ [.r,s.] by A8, XBOOLE_0:def 5; :: thesis: verum