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

proof

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

end;

assume t in [.r,q.] ; :: thesis: t in [.r,s.[ /\ [.p,q.]
then A3: ( r <= t & t <= q ) by Th1;
then ( t < s & p <= t ) 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