let r, s, t be ext-real number ; :: 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 ext-real number ; :: 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 ( ( r <= p & p <= s ) or ( s <= p & p <= t ) ) by Th1;
then ( r <= p & p <= t ) by A1, A2, XXREAL_0:2;
hence p in [.r,t.] by 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;
hence p in [.r,s.] \/ [.s,t.] by XBOOLE_0:def 3; :: thesis: verum