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 A1: t in [.r,s.[ /\ [.p,q.[ ; :: thesis: t in [.(max r,p),(min s,q).[
then A2: t in [.r,s.[ by XBOOLE_0:def 4;
A3: t in [.p,q.[ by A1, XBOOLE_0:def 4;
A4: r <= t by A2, Th3;
A5: t < s by A2, Th3;
A6: p <= t by A3, Th3;
A7: t < q by A3, Th3;
A8: max r,p <= t by A4, A6, XXREAL_0:28;
t < min s,q by A5, A7, XXREAL_0:21;
hence t in [.(max r,p),(min s,q).[ by A8, Th3; :: thesis: verum

end;

assume A9: t in [.(max r,p),(min s,q).[ ; :: thesis: t in [.r,s.[ /\ [.p,q.[
then A10: max r,p <= t by Th3;
A11: t < min s,q by A9, Th3;
A12: r <= t by A10, XXREAL_0:30;
A13: p <= t by A10, XXREAL_0:30;
A14: t < s by A11, XXREAL_0:23;
A15: t < q by A11, XXREAL_0:23;
A16: t in [.r,s.[ by A12, A14, Th3;
t in [.p,q.[ by A13, A15, Th3;
hence t in [.r,s.[ /\ [.p,q.[ by A16, XBOOLE_0:def 4; :: thesis: verum