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 Th3;
then ( max r,p <= t & t < min s,q ) by XXREAL_0:21, XXREAL_0:28;
hence t in [.(max r,p),(min s,q).[ by Th3; :: 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 Th3;
then ( r <= t & p <= t & t < s & t < q ) by XXREAL_0:23, XXREAL_0:30;
then ( t in [.r,s.[ & t in [.p,q.[ ) by Th3;
hence t in [.r,s.[ /\ [.p,q.[ by XBOOLE_0:def 4; :: thesis: verum