let X be set ; for f being PartFunc of REAL,REAL st f | X is uniformly_continuous holds
(abs f) | X is uniformly_continuous
let f be PartFunc of REAL,REAL; ( f | X is uniformly_continuous implies (abs f) | X is uniformly_continuous )
assume A1:
f | X is uniformly_continuous
; (abs f) | X is uniformly_continuous
now for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1, x2 being Real st x1 in dom ((abs f) | X) & x2 in dom ((abs f) | X) & |.(x1 - x2).| < s holds
|.(((abs f) . x1) - ((abs f) . x2)).| < r ) )let r be
Real;
( 0 < r implies ex s being Real st
( 0 < s & ( for x1, x2 being Real st x1 in dom ((abs f) | X) & x2 in dom ((abs f) | X) & |.(x1 - x2).| < s holds
|.(((abs f) . x1) - ((abs f) . x2)).| < r ) ) )assume
0 < r
;
ex s being Real st
( 0 < s & ( for x1, x2 being Real st x1 in dom ((abs f) | X) & x2 in dom ((abs f) | X) & |.(x1 - x2).| < s holds
|.(((abs f) . x1) - ((abs f) . x2)).| < r ) )then consider s being
Real such that A2:
0 < s
and A3:
for
x1,
x2 being
Real st
x1 in dom (f | X) &
x2 in dom (f | X) &
|.(x1 - x2).| < s holds
|.((f . x1) - (f . x2)).| < r
by A1, Th1;
take s =
s;
( 0 < s & ( for x1, x2 being Real st x1 in dom ((abs f) | X) & x2 in dom ((abs f) | X) & |.(x1 - x2).| < s holds
|.(((abs f) . x1) - ((abs f) . x2)).| < r ) )thus
0 < s
by A2;
for x1, x2 being Real st x1 in dom ((abs f) | X) & x2 in dom ((abs f) | X) & |.(x1 - x2).| < s holds
|.(((abs f) . x1) - ((abs f) . x2)).| < rlet x1,
x2 be
Real;
( x1 in dom ((abs f) | X) & x2 in dom ((abs f) | X) & |.(x1 - x2).| < s implies |.(((abs f) . x1) - ((abs f) . x2)).| < r )assume that A4:
x1 in dom ((abs f) | X)
and A5:
x2 in dom ((abs f) | X)
and A6:
|.(x1 - x2).| < s
;
|.(((abs f) . x1) - ((abs f) . x2)).| < r
x2 in dom (abs f)
by A5, RELAT_1:57;
then A7:
x2 in dom f
by VALUED_1:def 11;
x2 in X
by A5, RELAT_1:57;
then A8:
x2 in dom (f | X)
by A7, RELAT_1:57;
|.(((abs f) . x1) - ((abs f) . x2)).| =
|.(|.(f . x1).| - ((abs f) . x2)).|
by VALUED_1:18
.=
|.(|.(f . x1).| - |.(f . x2).|).|
by VALUED_1:18
;
then A9:
|.(((abs f) . x1) - ((abs f) . x2)).| <= |.((f . x1) - (f . x2)).|
by COMPLEX1:64;
x1 in dom (abs f)
by A4, RELAT_1:57;
then A10:
x1 in dom f
by VALUED_1:def 11;
x1 in X
by A4, RELAT_1:57;
then
x1 in dom (f | X)
by A10, RELAT_1:57;
then
|.((f . x1) - (f . x2)).| < r
by A3, A6, A8;
hence
|.(((abs f) . x1) - ((abs f) . x2)).| < r
by A9, XXREAL_0:2;
verum end;
hence
(abs f) | X is uniformly_continuous
by Th1; verum