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 Th4; :: thesis: verum

end;

assume p in ].r,s.[ ; :: thesis: p in ].r,t.[ \ [.s,t.[
then ( r < p & p < s ) by Th4;
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