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

proof

assume p in [.r,s.] \/ ].s,t.[ ; :: thesis: p in [.r,t.[
then ( p in [.r,s.] or p in ].s,t.[ ) by XBOOLE_0:def 3;
then A3: ( ( r <= p & p <= s ) or ( s < p & p < t ) ) by Th1, Th4;
then A4: r <= p by A1, XXREAL_0:2;
p < t by A2, A3, XXREAL_0:2;
hence p in [.r,t.[ by A4, Th3; :: thesis: verum

end;

assume p in [.r,t.[ ; :: thesis: p in [.r,s.] \/ ].s,t.[
then ( ( r <= p & p <= s ) or ( s < p & p < t ) ) by Th3;
then ( p in [.r,s.] or p in ].s,t.[ ) by Th1, Th4;
hence p in [.r,s.] \/ ].s,t.[ by XBOOLE_0:def 3; :: thesis: verum