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, Th2;
then A4: r <= p by A1, XXREAL_0:2;
p <= t by A2, A3, XXREAL_0:2;
hence p in [.r,t.] by A4, Th1; :: 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 Th1;
then ( p in [.r,s.] or p in ].s,t.] ) by Th1, Th2;
hence p in [.r,s.] \/ ].s,t.] by XBOOLE_0:def 3; :: thesis: verum