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 p in [.r,t.[ \ ].s,t.[ ; :: thesis: p in [.r,s.]
then ( p in [.r,t.[ & not p in ].s,t.[ ) by XBOOLE_0:def 5;
then ( r <= p & p < t & ( p <= s or t <= p ) ) by Th3, Th4;
hence p in [.r,s.] by Th1; :: thesis: verum

end;

assume p in [.r,s.] ; :: thesis: p in [.r,t.[ \ ].s,t.[
then ( r <= p & p <= s ) by Th1;
then ( r <= p & p < t & ( p <= s or t <= p ) ) by A1, XXREAL_0:2;
then ( p in [.r,t.[ & not p in ].s,t.[ ) by Th3, Th4;
hence p in [.r,t.[ \ ].s,t.[ by XBOOLE_0:def 5; :: thesis: verum