let p, q, r, s be ExtReal; :: 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 ExtReal; :: 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 A3: t in ].r,s.] /\ [.p,q.[ ; :: thesis: t in [.p,s.]
then A4: t in ].r,s.] by XBOOLE_0:def 4;
A5: t in [.p,q.[ by A3, XBOOLE_0:def 4;
A6: t <= s by A4, Th2;
p <= t by A5, Th3;
hence t in [.p,s.] by A6, Th1; :: thesis: verum

end;

assume A7: t in [.p,s.] ; :: thesis: t in ].r,s.] /\ [.p,q.[
then A8: p <= t by Th1;
A9: t <= s by A7, Th1;
A10: r < t by A1, A8, XXREAL_0:2;
A11: t < q by A2, A9, XXREAL_0:2;
A12: t in ].r,s.] by A9, A10, Th2;
t in [.p,q.[ by A8, A11, Th3;
hence t in ].r,s.] /\ [.p,q.[ by A12, XBOOLE_0:def 4; :: thesis: verum