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, Th3;
p <= t by A5, Th1;
hence t in [.p,s.[ by A6, Th3; :: thesis: verum

end;

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