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

proof

assume t in [.r,s.] /\ [.p,q.] ; :: thesis: t in [.(max r,p),(min s,q).]
then ( t in [.r,s.] & t in [.p,q.] ) by XBOOLE_0:def 4;
then ( r <= t & t <= s & p <= t & t <= q ) by Th1;
then ( max r,p <= t & t <= min s,q ) by XXREAL_0:20, XXREAL_0:28;
hence t in [.(max r,p),(min s,q).] by Th1; :: thesis: verum

end;

assume t in [.(max r,p),(min s,q).] ; :: thesis: t in [.r,s.] /\ [.p,q.]
then ( max r,p <= t & t <= min s,q ) by Th1;
then ( r <= t & p <= t & t <= s & t <= q ) by XXREAL_0:22, XXREAL_0:30;
then ( t in [.r,s.] & t in [.p,q.] ) by Th1;
hence t in [.r,s.] /\ [.p,q.] by XBOOLE_0:def 4; :: thesis: verum