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 Th2, Th3;
then ( r < p & p < t ) by A1, A2, XXREAL_0:2;
hence p in ].r,t.[ by Th4; :: 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 Th4;
then ( p in ].r,s.] or p in [.s,t.[ ) by Th3, Th2;
hence p in ].r,s.] \/ [.s,t.[ by XBOOLE_0:def 3; :: thesis: verum