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, Th3;
( p <= s or t <= p ) by A3, Th4;
hence p in [.r,s.] by A2, A4, Th1, Th3; :: thesis: verum

end;

assume A5: p in [.r,s.] ; :: thesis: p in [.r,t.[ \ ].s,t.[
then A6: p <= s by Th1;
A7: r <= p by A5, Th1;
p < t by A1, A6, XXREAL_0:2;
then A8: p in [.r,t.[ by A7, Th3;
not p in ].s,t.[ by A6, Th4;
hence p in [.r,t.[ \ ].s,t.[ by A8, XBOOLE_0:def 5; :: thesis: verum