let f be PartFunc of REAL,REAL; ( f is total & ( for r1, r2 being Real holds |.((f . r1) - (f . r2)).| <= (r1 - r2) ^2 ) implies ( f is_differentiable_on [#] REAL & f | ([#] REAL) is V8() ) )
assume that
A1:
f is total
and
A2:
for r1, r2 being Real holds |.((f . r1) - (f . r2)).| <= (r1 - r2) ^2
; ( f is_differentiable_on [#] REAL & f | ([#] REAL) is V8() )
A3:
dom f = [#] REAL
by A1, PARTFUN1:def 2;
A4:
now for r1, r2 being Element of REAL st r1 in ([#] REAL) /\ (dom f) & r2 in ([#] REAL) /\ (dom f) holds
f . r1 = f . r2let r1,
r2 be
Element of
REAL ;
( r1 in ([#] REAL) /\ (dom f) & r2 in ([#] REAL) /\ (dom f) implies f . r1 = f . r2 )assume that A5:
r1 in ([#] REAL) /\ (dom f)
and A6:
r2 in ([#] REAL) /\ (dom f)
;
f . r1 = f . r2set rx =
max (
r1,
r2);
set rn =
min (
r1,
r2);
A7:
r1 + 0 < (max (r1,r2)) + 1
by XREAL_1:8, XXREAL_0:25;
A8:
r2 + 0 < (max (r1,r2)) + 1
by XREAL_1:8, XXREAL_0:25;
(min (r1,r2)) - 1
< r2 - 0
by XREAL_1:15, XXREAL_0:17;
then
r2 in { g2 where g2 is Real : ( (min (r1,r2)) - 1 < g2 & g2 < (max (r1,r2)) + 1 ) }
by A8;
then A9:
r2 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[
by RCOMP_1:def 2;
r2 in dom f
by A6, XBOOLE_0:def 4;
then A10:
r2 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ /\ (dom f)
by A9, XBOOLE_0:def 4;
(min (r1,r2)) - 1
< r1 - 0
by XREAL_1:15, XXREAL_0:17;
then
r1 in { g1 where g1 is Real : ( (min (r1,r2)) - 1 < g1 & g1 < (max (r1,r2)) + 1 ) }
by A7;
then A11:
r1 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[
by RCOMP_1:def 2;
r1 in dom f
by A5, XBOOLE_0:def 4;
then A12:
r1 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ /\ (dom f)
by A11, XBOOLE_0:def 4;
for
g1,
g2 being
Real st
g1 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ &
g2 in ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ holds
|.((f . g1) - (f . g2)).| <= (g1 - g2) ^2
by A2;
then
f | ].((min (r1,r2)) - 1),((max (r1,r2)) + 1).[ is
V8()
by A3, Th25;
hence
f . r1 = f . r2
by A12, A10, PARTFUN2:58;
verum end;
for r1, r2 being Real st r1 in [#] REAL & r2 in [#] REAL holds
|.((f . r1) - (f . r2)).| <= (r1 - r2) ^2
by A2;
hence
( f is_differentiable_on [#] REAL & f | ([#] REAL) is V8() )
by A3, A4, Th24, PARTFUN2:58; verum