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
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
is_uniformly_continuous_on X

let f be PartFunc of S,T; :: thesis:
assume A1: f is_uniformly_continuous_on X ; :: thesis:
then X c= dom f ;
hence A2: X c= dom 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
|.(( /. x1) - ( /. 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
|.(( /. x1) - ( /. 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
|.(( /. x1) - ( /. 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
|.(( /. x1) - ( /. 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
|.(( /. x1) - ( /. x2)).| < r

let x1, x2 be Point of S; :: thesis: ( x1 in X & x2 in X & ||.(x1 - x2).|| < s implies |.(( /. x1) - ( /. x2)).| < r )
assume that
A5: x1 in X and
A6: x2 in X and
A7: ||.(x1 - x2).|| < s ; :: thesis: |.(( /. x1) - ( /. x2)).| < r
|.(( /. x1) - ( /. x2)).| = |.(( . x1) - ( /. x2)).| by
.= |.(( . x1) - ( . x2)).| by
.= |.(||.(f /. x1).|| - ( . x2)).| by
.= |.(||.(f /. x1).|| - ||.(f /. x2).||).| by ;
then A8: |.(( /. x1) - ( /. x2)).| <= ||.((f /. x1) - (f /. x2)).|| by NORMSP_1:9;
||.((f /. x1) - (f /. x2)).|| < r by A4, A5, A6, A7;
hence |.(( /. x1) - ( /. x2)).| < r by ; :: thesis: verum