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

proof

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

end;

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