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

proof

assume t in [.r,s.] ; :: thesis: t in {r} \/ ].r,s.]
then ( t in ].r,s.] or t = r ) by Th6;
then ( t in ].r,s.] or t in {r} ) by TARSKI:def 1;
hence t in {r} \/ ].r,s.] by XBOOLE_0:def 3; :: thesis: verum

end;

assume t in {r} \/ ].r,s.] ; :: thesis: t in [.r,s.]
then ( t in ].r,s.] or t in {r} ) by XBOOLE_0:def 3;
then ( t in ].r,s.] or t = r ) by TARSKI:def 1;
hence t in [.r,s.] by A1, Th1, Th12; :: thesis: verum