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

end;

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