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 Th2;
then ( max r,p < t & t <= min s,q ) by XXREAL_0:20, XXREAL_0:29;
hence t in ].(max r,p),(min s,q).] by Th2; :: 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 Th2;
then ( r < t & p < t & t <= s & t <= q ) by XXREAL_0:22, XXREAL_0:31;
then ( t in ].r,s.] & t in ].p,q.] ) by Th2;
hence t in ].r,s.] /\ ].p,q.] by XBOOLE_0:def 4; :: thesis: verum