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 Th3, Th4;
hence t in [.p,s.[ by Th3; :: thesis: verum

end;

assume t in [.p,s.[ ; :: thesis: t in ].r,s.[ /\ [.p,q.[
then A3: ( p <= t & t < s ) by Th3;
then ( r < t & t < q ) by A1, A2, XXREAL_0:2;
then ( t in ].r,s.[ & t in [.p,q.[ ) by A3, Th3, Th4;
hence t in ].r,s.[ /\ [.p,q.[ by XBOOLE_0:def 4; :: thesis: verum