let r, s, t be ext-real number ; :: thesis: ( r < s & s < t implies ].r,t.[ \ {s} = ].r,s.[ \/ ].s,t.[ )
assume that
A1: r < s and
A2: s < t ; :: thesis: ].r,t.[ \ {s} = ].r,s.[ \/ ].s,t.[
let p be ext-real number ; :: according to MEMBERED:def 14 :: thesis: ( ( not p in ].r,t.[ \ {s} or p in ].r,s.[ \/ ].s,t.[ ) & ( not p in ].r,s.[ \/ ].s,t.[ or p in ].r,t.[ \ {s} ) )
thus ( p in ].r,t.[ \ {s} implies p in ].r,s.[ \/ ].s,t.[ ) :: thesis: ( not p in ].r,s.[ \/ ].s,t.[ or p in ].r,t.[ \ {s} )

proof

assume A3: p in ].r,t.[ \ {s} ; :: thesis: p in ].r,s.[ \/ ].s,t.[
then not p in {s} by XBOOLE_0:def 5;
then p <> s by TARSKI:def 1;
then ( ( r < p & p < s ) or ( s < p & p < t ) ) by A3, Th4, XXREAL_0:1;
then ( p in ].r,s.[ or p in ].s,t.[ ) by Th4;
hence p in ].r,s.[ \/ ].s,t.[ by XBOOLE_0:def 3; :: thesis: verum

end;

assume p in ].r,s.[ \/ ].s,t.[ ; :: thesis: p in ].r,t.[ \ {s}
then ( p in ].r,s.[ or p in ].s,t.[ ) by XBOOLE_0:def 3;
then ( ( r < p & p < s ) or ( s < p & p < t ) ) by Th4;
then ( r < p & p < t & p <> s ) by A1, A2, XXREAL_0:2;
then ( p in ].r,t.[ & not p in {s} ) by Th4, TARSKI:def 1;
hence p in ].r,t.[ \ {s} by XBOOLE_0:def 5; :: thesis: verum