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 Th3;
hence t in [.r,q.[ by Th3; :: thesis: verum

end;

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