let X be set ; :: thesis: 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; :: thesis: ( f | X is uniformly_continuous implies (abs f) | X is uniformly_continuous )
assume A1: f | X is uniformly_continuous ; :: thesis: (abs f) | X is uniformly_continuous

now :: thesis: 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; :: thesis: ( 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 ; :: thesis: 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; :: thesis: ( 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; :: thesis: 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 x1, x2 be Real; :: thesis: ( 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 ; :: thesis: |.(((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; :: thesis: verum

end;

hence (abs f) | X is uniformly_continuous by Th1; :: thesis: verum