let p, q, r, s be ExtReal; ( r >= p & s >= q implies [.r,s.[ /\ [.p,q.[ = [.r,q.[ )
assume that
A1:
r >= p
and
A2:
s >= q
; [.r,s.[ /\ [.p,q.[ = [.r,q.[
let t be ExtReal; MEMBERED:def 14 ( ( not t in [.r,s.[ /\ [.p,q.[ or t in [.r,q.[ ) & ( not t in [.r,q.[ or t in [.r,s.[ /\ [.p,q.[ ) )
thus
( t in [.r,s.[ /\ [.p,q.[ implies t in [.r,q.[ )
( not t in [.r,q.[ or t in [.r,s.[ /\ [.p,q.[ )proof
assume A3:
t in [.r,s.[ /\ [.p,q.[
;
t in [.r,q.[
then A4:
t in [.r,s.[
by XBOOLE_0:def 4;
A5:
t in [.p,q.[
by A3, XBOOLE_0:def 4;
A6:
r <= t
by A4, Th3;
t < q
by A5, Th3;
hence
t in [.r,q.[
by A6, Th3;
verum
end;
assume A7:
t in [.r,q.[
; t in [.r,s.[ /\ [.p,q.[
then A8:
r <= t
by Th3;
A9:
t < q
by A7, Th3;
then A10:
t < s
by A2, XXREAL_0:2;
A11:
p <= t
by A1, A8, XXREAL_0:2;
A12:
t in [.r,s.[
by A8, A10, Th3;
t in [.p,q.[
by A9, A11, Th3;
hence
t in [.r,s.[ /\ [.p,q.[
by A12, XBOOLE_0:def 4; verum