let r, p, s, q be ext-real number ; :: thesis: ( r <= p & s <= q implies [.r,s.[ /\ [.p,q.] = [.p,s.[ )
assume that
A1: r <= p and
A2: s <= q ; :: thesis: [.r,s.[ /\ [.p,q.] = [.p,s.[
let t be ext-real number ; :: according to MEMBERED:def 14 :: thesis: ( ( not t in [.r,s.[ /\ [.p,q.] or t in [.p,s.[ ) & ( not t in [.p,s.[ or t in [.r,s.[ /\ [.p,q.] ) )
thus ( t in [.r,s.[ /\ [.p,q.] implies t in [.p,s.[ ) :: thesis: ( not t in [.p,s.[ or t in [.r,s.[ /\ [.p,q.] )

proof

assume t in [.r,s.[ /\ [.p,q.] ; :: thesis: t in [.p,s.[
then ( t in [.r,s.[ & t in [.p,q.] ) by XBOOLE_0:def 4;
then ( t < s & p <= t ) by Th1, Th3;
hence t in [.p,s.[ by Th3; :: thesis: verum

end;

assume t in [.p,s.[ ; :: thesis: t in [.r,s.[ /\ [.p,q.]
then A3: ( p <= t & t < s ) by Th3;
then ( r <= t & t <= q ) by A1, A2, XXREAL_0:2;
then ( t in [.r,s.[ & t in [.p,q.] ) by A3, Th1, Th3;
hence t in [.r,s.[ /\ [.p,q.] by XBOOLE_0:def 4; :: thesis: verum