let r, s, t be ext-real number ; :: thesis: ( r < s implies ].r,t.[ \ ].r,s.[ = [.s,t.[ )
assume A1: r < s ; :: thesis: ].r,t.[ \ ].r,s.[ = [.s,t.[
let p be ext-real number ; :: 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, Th4;
( p <= r or s <= p ) by A3, Th4;
hence p in [.s,t.[ by A2, A4, Th3, Th4; :: thesis: verum

end;

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