let p, q, r, s be ExtReal; ( r < p & s < q implies ].r,s.] /\ [.p,q.[ = [.p,s.] )
assume that
A1:
r < p
and
A2:
s < q
; ].r,s.] /\ [.p,q.[ = [.p,s.]
let t be ExtReal; MEMBERED:def 14 ( ( 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.] )
( not t in [.p,s.] or t in ].r,s.] /\ [.p,q.[ )proof
assume A3:
t in ].r,s.] /\ [.p,q.[
;
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;
verum
end;
assume A7:
t in [.p,s.]
; 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; verum