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