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

proof

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

end;

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