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 Th4;
then ( max r,p < t & t < min s,q ) by XXREAL_0:21, XXREAL_0:29;
hence t in ].(max r,p),(min s,q).[ by Th4; :: 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 Th4;
then ( r < t & p < t & t < s & t < q ) by XXREAL_0:23, XXREAL_0:31;
then ( t in ].r,s.[ & t in ].p,q.[ ) by Th4;
hence t in ].r,s.[ /\ ].p,q.[ by XBOOLE_0:def 4; :: thesis: verum