let p, g be Real; for f being PartFunc of REAL,REAL st ].p,g.[ c= dom f & f is_differentiable_on ].p,g.[ & ( for x being Real st x in ].p,g.[ holds
diff (f,x) < 0 ) holds
f | ].p,g.[ is decreasing
let f be PartFunc of REAL,REAL; ( ].p,g.[ c= dom f & f is_differentiable_on ].p,g.[ & ( for x being Real st x in ].p,g.[ holds
diff (f,x) < 0 ) implies f | ].p,g.[ is decreasing )
assume that
A1:
].p,g.[ c= dom f
and
A2:
f is_differentiable_on ].p,g.[
and
A3:
for x being Real st x in ].p,g.[ holds
diff (f,x) < 0
; f | ].p,g.[ is decreasing
now for x1, x2 being Real st x1 in ].p,g.[ /\ (dom f) & x2 in ].p,g.[ /\ (dom f) & x1 < x2 holds
f . x2 < f . x1let x1,
x2 be
Real;
( x1 in ].p,g.[ /\ (dom f) & x2 in ].p,g.[ /\ (dom f) & x1 < x2 implies f . x2 < f . x1 )assume that A4:
(
x1 in ].p,g.[ /\ (dom f) &
x2 in ].p,g.[ /\ (dom f) )
and A5:
x1 < x2
;
f . x2 < f . x1A6:
0 < x2 - x1
by A5, XREAL_1:50;
reconsider Z =
].x1,x2.[ as
open Subset of
REAL ;
(
x1 in ].p,g.[ &
x2 in ].p,g.[ )
by A4, XBOOLE_0:def 4;
then A7:
[.x1,x2.] c= ].p,g.[
by XXREAL_2:def 12;
f | ].p,g.[ is
continuous
by A2, FDIFF_1:25;
then A8:
f | [.x1,x2.] is
continuous
by A7, FCONT_1:16;
A9:
Z c= [.x1,x2.]
by XXREAL_1:25;
then
f is_differentiable_on Z
by A2, A7, FDIFF_1:26, XBOOLE_1:1;
then A10:
ex
x0 being
Real st
(
x0 in ].x1,x2.[ &
diff (
f,
x0)
= ((f . x2) - (f . x1)) / (x2 - x1) )
by A1, A5, A7, A8, Th3, XBOOLE_1:1;
Z c= ].p,g.[
by A7, A9;
then
(f . x2) - (f . x1) < 0
by A3, A6, A10;
then
f . x2 < (f . x1) + 0
by XREAL_1:19;
hence
f . x2 < f . x1
;
verum end;
hence
f | ].p,g.[ is decreasing
by RFUNCT_2:21; verum