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 Th2, Th3;
hence t in [.p,s.] by Th1; :: thesis: verum

end;

assume t in [.p,s.] ; :: thesis: t in ].r,s.] /\ [.p,q.[
then A3: ( p <= t & t <= s ) by Th1;
then ( r < t & t < q ) by A1, A2, XXREAL_0:2;
then ( t in ].r,s.] & t in [.p,q.[ ) by A3, Th2, Th3;
hence t in ].r,s.] /\ [.p,q.[ by XBOOLE_0:def 4; :: thesis: verum