let X be set ; :: thesis: for S, T being RealNormSpace

for f being PartFunc of S,T st f is_uniformly_continuous_on X holds

||.f.|| is_uniformly_continuous_on X

let S, T be RealNormSpace; :: thesis: for f being PartFunc of S,T st f is_uniformly_continuous_on X holds

||.f.|| is_uniformly_continuous_on X

let f be PartFunc of S,T; :: thesis: ( f is_uniformly_continuous_on X implies ||.f.|| is_uniformly_continuous_on X )

assume A1: f is_uniformly_continuous_on X ; :: thesis: ||.f.|| is_uniformly_continuous_on X

then X c= dom f ;

hence A2: X c= dom ||.f.|| by NORMSP_0:def 3; :: according to NFCONT_2:def 2 :: thesis: for r being Real st 0 < r holds

ex s being Real st

( 0 < s & ( for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds

|.((||.f.|| /. x1) - (||.f.|| /. x2)).| < r ) )

let r be Real; :: thesis: ( 0 < r implies ex s being Real st

( 0 < s & ( for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds

|.((||.f.|| /. x1) - (||.f.|| /. x2)).| < r ) ) )

assume 0 < r ; :: thesis: ex s being Real st

( 0 < s & ( for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds

|.((||.f.|| /. x1) - (||.f.|| /. x2)).| < r ) )

then consider s being Real such that

A3: 0 < s and

A4: for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds

||.((f /. x1) - (f /. x2)).|| < r by A1;

take s ; :: thesis: ( 0 < s & ( for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds

|.((||.f.|| /. x1) - (||.f.|| /. x2)).| < r ) )

thus 0 < s by A3; :: thesis: for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds

|.((||.f.|| /. x1) - (||.f.|| /. x2)).| < r

let x1, x2 be Point of S; :: thesis: ( x1 in X & x2 in X & ||.(x1 - x2).|| < s implies |.((||.f.|| /. x1) - (||.f.|| /. x2)).| < r )

assume that

A5: x1 in X and

A6: x2 in X and

A7: ||.(x1 - x2).|| < s ; :: thesis: |.((||.f.|| /. x1) - (||.f.|| /. x2)).| < r

|.((||.f.|| /. x1) - (||.f.|| /. x2)).| = |.((||.f.|| . x1) - (||.f.|| /. x2)).| by A2, A5, PARTFUN1:def 6

.= |.((||.f.|| . x1) - (||.f.|| . x2)).| by A2, A6, PARTFUN1:def 6

.= |.(||.(f /. x1).|| - (||.f.|| . x2)).| by A2, A5, NORMSP_0:def 3

.= |.(||.(f /. x1).|| - ||.(f /. x2).||).| by A2, A6, NORMSP_0:def 3 ;

then A8: |.((||.f.|| /. x1) - (||.f.|| /. x2)).| <= ||.((f /. x1) - (f /. x2)).|| by NORMSP_1:9;

||.((f /. x1) - (f /. x2)).|| < r by A4, A5, A6, A7;

hence |.((||.f.|| /. x1) - (||.f.|| /. x2)).| < r by A8, XXREAL_0:2; :: thesis: verum

for f being PartFunc of S,T st f is_uniformly_continuous_on X holds

||.f.|| is_uniformly_continuous_on X

let S, T be RealNormSpace; :: thesis: for f being PartFunc of S,T st f is_uniformly_continuous_on X holds

||.f.|| is_uniformly_continuous_on X

let f be PartFunc of S,T; :: thesis: ( f is_uniformly_continuous_on X implies ||.f.|| is_uniformly_continuous_on X )

assume A1: f is_uniformly_continuous_on X ; :: thesis: ||.f.|| is_uniformly_continuous_on X

then X c= dom f ;

hence A2: X c= dom ||.f.|| by NORMSP_0:def 3; :: according to NFCONT_2:def 2 :: thesis: for r being Real st 0 < r holds

ex s being Real st

( 0 < s & ( for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds

|.((||.f.|| /. x1) - (||.f.|| /. x2)).| < r ) )

let r be Real; :: thesis: ( 0 < r implies ex s being Real st

( 0 < s & ( for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds

|.((||.f.|| /. x1) - (||.f.|| /. x2)).| < r ) ) )

assume 0 < r ; :: thesis: ex s being Real st

( 0 < s & ( for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds

|.((||.f.|| /. x1) - (||.f.|| /. x2)).| < r ) )

then consider s being Real such that

A3: 0 < s and

A4: for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds

||.((f /. x1) - (f /. x2)).|| < r by A1;

take s ; :: thesis: ( 0 < s & ( for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds

|.((||.f.|| /. x1) - (||.f.|| /. x2)).| < r ) )

thus 0 < s by A3; :: thesis: for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds

|.((||.f.|| /. x1) - (||.f.|| /. x2)).| < r

let x1, x2 be Point of S; :: thesis: ( x1 in X & x2 in X & ||.(x1 - x2).|| < s implies |.((||.f.|| /. x1) - (||.f.|| /. x2)).| < r )

assume that

A5: x1 in X and

A6: x2 in X and

A7: ||.(x1 - x2).|| < s ; :: thesis: |.((||.f.|| /. x1) - (||.f.|| /. x2)).| < r

|.((||.f.|| /. x1) - (||.f.|| /. x2)).| = |.((||.f.|| . x1) - (||.f.|| /. x2)).| by A2, A5, PARTFUN1:def 6

.= |.((||.f.|| . x1) - (||.f.|| . x2)).| by A2, A6, PARTFUN1:def 6

.= |.(||.(f /. x1).|| - (||.f.|| . x2)).| by A2, A5, NORMSP_0:def 3

.= |.(||.(f /. x1).|| - ||.(f /. x2).||).| by A2, A6, NORMSP_0:def 3 ;

then A8: |.((||.f.|| /. x1) - (||.f.|| /. x2)).| <= ||.((f /. x1) - (f /. x2)).|| by NORMSP_1:9;

||.((f /. x1) - (f /. x2)).|| < r by A4, A5, A6, A7;

hence |.((||.f.|| /. x1) - (||.f.|| /. x2)).| < r by A8, XXREAL_0:2; :: thesis: verum