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 A2: p in [.r,t.] \ [.r,s.[ ; :: thesis: p in [.s,t.]
then A3: not p in [.r,s.[ by XBOOLE_0:def 5;
A4: p <= t by A2, Th1;
( p < r or s <= p ) by A3, Th3;
hence p in [.s,t.] by A2, A4, Th1; :: thesis: verum

end;

assume A5: p in [.s,t.] ; :: thesis: p in [.r,t.] \ [.r,s.[
then A6: s <= p by Th1;
then A7: r <= p by A1, XXREAL_0:2;
p <= t by A5, Th1;
then A8: p in [.r,t.] by A7, Th1;
not p in [.r,s.[ by A6, Th3;
hence p in [.r,t.] \ [.r,s.[ by A8, XBOOLE_0:def 5; :: thesis: verum