let r, s, t be ExtReal; :: 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 ExtReal; :: 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, Th3, XXREAL_0:1;
then ( p in [.r,s.[ or p in ].s,t.[ ) by Th3, 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 A4: ( ( r <= p & p < s ) or ( s < p & p < t ) ) by Th3, Th4;
then A5: r <= p by A1, XXREAL_0:2;
p < t by A2, A4, XXREAL_0:2;
then A6: p in [.r,t.[ by A5, Th3;
not p in {s} by A4, TARSKI:def 1;
hence p in [.r,t.[ \ {s} by A6, XBOOLE_0:def 5; :: thesis: verum