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, Th4;
hence t in ].r,q.] by Th2; :: thesis: verum

end;

assume t in ].r,q.] ; :: thesis: t in ].r,s.[ /\ [.p,q.]
then A3: ( r < t & t <= q ) by Th2;
then ( t < s & p <= t ) by A1, A2, XXREAL_0:2;
then ( t in ].r,s.[ & t in [.p,q.] ) by A3, Th1, Th4;
hence t in ].r,s.[ /\ [.p,q.] by XBOOLE_0:def 4; :: thesis: verum