let r, s be ExtReal; :: thesis: ( r < s implies [.r,s.[ = {r} \/ ].r,s.[ )
assume A1: r < s ; :: thesis: [.r,s.[ = {r} \/ ].r,s.[
let t be ExtReal; :: 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 Th8;
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, Th3, Th14; :: thesis: verum