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

(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 ) )

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