let r, s, t be ExtReal; :: thesis: ( s <= t implies ].r,t.] \ [.s,t.] = ].r,s.[ )
assume A1: s <= t ; :: thesis: ].r,t.] \ [.s,t.] = ].r,s.[
let p be ExtReal; :: 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, Th1;
hence p in ].r,s.[ by A2, A4, Th2, Th4; :: thesis: verum

end;

assume A5: p in ].r,s.[ ; :: thesis: p in ].r,t.] \ [.s,t.]
then A6: p < s by Th4;
A7: r < p by A5, Th4;
p <= t by A1, A6, XXREAL_0:2;
then A8: p in ].r,t.] by A7, Th2;
not p in [.s,t.] by A6, Th1;
hence p in ].r,t.] \ [.s,t.] by A8, XBOOLE_0:def 5; :: thesis: verum