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

proof

assume t in [.r,s.] ; :: thesis: t in [.r,s.[ \/ {s}
then ( t in [.r,s.[ or t = s ) by Th7;
then ( t in [.r,s.[ or t in {s} ) by TARSKI:def 1;
hence t in [.r,s.[ \/ {s} by XBOOLE_0:def 3; :: thesis: verum

end;

assume t in [.r,s.[ \/ {s} ; :: thesis: t in [.r,s.]
then ( t in [.r,s.[ or t in {s} ) by XBOOLE_0:def 3;
then ( t in [.r,s.[ or t = s ) by TARSKI:def 1;
hence t in [.r,s.] by A1, Th1, Th13; :: thesis: verum