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

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) & abs (x1 - x2) < s holds
abs (((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) & abs (x1 - x2) < s holds
abs (((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) & abs (x1 - x2) < s holds
abs ((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) & abs (x1 - x2) < s holds
abs (((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) & abs (x1 - x2) < s holds
abs (((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) & abs (x1 - x2) < s implies abs (((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: abs (x1 - x2) < s ; :: thesis: abs (((abs f) . x1) - ((abs f) . x2)) < r
x2 in dom (abs f) by A5, RELAT_1:86;
then A7: x2 in dom f by VALUED_1:def 11;
x2 in X by A5, RELAT_1:86;
then A8: x2 in dom (f | X) by A7, RELAT_1:86;
abs (((abs f) . x1) - ((abs f) . x2)) = abs ((abs (f . x1)) - ((abs f) . x2)) by VALUED_1:18
.= abs ((abs (f . x1)) - (abs (f . x2))) by VALUED_1:18 ;
then A9: abs (((abs f) . x1) - ((abs f) . x2)) <= abs ((f . x1) - (f . x2)) by COMPLEX1:150;
x1 in dom (abs f) by A4, RELAT_1:86;
then A10: x1 in dom f by VALUED_1:def 11;
x1 in X by A4, RELAT_1:86;
then x1 in dom (f | X) by A10, RELAT_1:86;
then abs ((f . x1) - (f . x2)) < r by A3, A6, A8;
hence abs (((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