let r, s be ext-real number ; :: thesis: ( r <= s implies [.r,s.] = ].r,s.[ \/ {r,s} )
assume A1: r <= s ; :: thesis: [.r,s.] = ].r,s.[ \/ {r,s}
let t be ext-real number ; :: according to MEMBERED:def 14 :: thesis: ( ( not t in [.r,s.] or t in ].r,s.[ \/ {r,s} ) & ( not t in ].r,s.[ \/ {r,s} or t in [.r,s.] ) )
thus ( t in [.r,s.] implies t in ].r,s.[ \/ {r,s} ) :: thesis: ( not t in ].r,s.[ \/ {r,s} or t in [.r,s.] )

proof

assume t in [.r,s.] ; :: thesis: t in ].r,s.[ \/ {r,s}
then ( t in ].r,s.[ or t = r or t = s ) by Th5;
then ( t in ].r,s.[ or t in {r,s} ) by TARSKI:def 2;
hence t in ].r,s.[ \/ {r,s} by XBOOLE_0:def 3; :: thesis: verum

end;

assume t in ].r,s.[ \/ {r,s} ; :: thesis: t in [.r,s.]
then ( t in ].r,s.[ or t in {r,s} ) by XBOOLE_0:def 3;
then ( t in ].r,s.[ or t = r or t = s ) by TARSKI:def 2;
hence t in [.r,s.] by A1, Th1, Th16; :: thesis: verum