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

proof

assume p in [.r,t.] \ [.r,s.[ ; :: thesis: p in [.s,t.]
then ( p in [.r,t.] & not p in [.r,s.[ ) by XBOOLE_0:def 5;
then ( r <= p & p <= t & ( p < r or s <= p ) ) by Th1, Th3;
hence p in [.s,t.] by Th1; :: thesis: verum

end;

assume p in [.s,t.] ; :: thesis: p in [.r,t.] \ [.r,s.[
then ( s <= p & p <= t ) by Th1;
then ( r <= p & p <= t & ( p < r or s <= p ) ) by A1, XXREAL_0:2;
then ( p in [.r,t.] & not p in [.r,s.[ ) by Th1, Th3;
hence p in [.r,t.] \ [.r,s.[ by XBOOLE_0:def 5; :: thesis: verum