let X, X1 be set ; :: thesis: for S, T being RealNormSpace
for f being PartFunc of S,T st f is_uniformly_continuous_on X & X1 c= X holds
f is_uniformly_continuous_on X1

let S, T be RealNormSpace; :: thesis: for f being PartFunc of S,T st f is_uniformly_continuous_on X & X1 c= X holds
f is_uniformly_continuous_on X1

let f be PartFunc of S,T; :: thesis: ( f is_uniformly_continuous_on X & X1 c= X implies f is_uniformly_continuous_on X1 )
assume that
A1: f is_uniformly_continuous_on X and
A2: X1 c= X ; :: thesis:
X c= dom f by A1;
hence X1 c= dom f by ; :: according to NFCONT_2:def 1 :: 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 X1 & x2 in X1 & ||.(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 X1 & x2 in X1 & ||.(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 X1 & x2 in X1 & ||.(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 X1 & x2 in X1 & ||.(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 X1 & x2 in X1 & ||.(x1 - x2).|| < s holds
||.((f /. x1) - (f /. x2)).|| < r

let x1, x2 be Point of S; :: thesis: ( x1 in X1 & x2 in X1 & ||.(x1 - x2).|| < s implies ||.((f /. x1) - (f /. x2)).|| < r )
assume ( x1 in X1 & x2 in X1 & ||.(x1 - x2).|| < s ) ; :: thesis: ||.((f /. x1) - (f /. x2)).|| < r
hence ||.((f /. x1) - (f /. x2)).|| < r by A2, A4; :: thesis: verum