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, Th4;
p <= t
by A5, Th3;
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, Th4;
t in [.p,q.[
by A8, A11, Th3;
hence
t in ].r,s.[ /\ [.p,q.[
by A12, XBOOLE_0:def 4; verum