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, Th3;
p <= t
by A5, Th1;
hence
t in [.p,s.[
by A6, Th3;
verum
end;
assume A7:
t in [.p,s.[
; 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; verum