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, Th4;
( p <= s or t <= p ) by A3, Th4;
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 Th2;
A7: r < p by A5, Th2;
p < t by A1, A6, XXREAL_0:2;
then A8: p in ].r,t.[ by A7, Th4;
not p in ].s,t.[ by A6, Th4;
hence p in ].r,t.[ \ ].s,t.[ by A8, XBOOLE_0:def 5; :: thesis: verum