let X be set ; :: thesis: for S being RealNormSpace
for f being PartFunc of the carrier of S,REAL st f is_uniformly_continuous_on X holds
f is_continuous_on X

let S be RealNormSpace; :: thesis: for f being PartFunc of the carrier of S,REAL st f is_uniformly_continuous_on X holds
f is_continuous_on X

let f be PartFunc of the carrier of S,REAL; :: thesis: ( f is_uniformly_continuous_on X implies f is_continuous_on X )
assume A1: f is_uniformly_continuous_on X ; :: thesis:
A2: now :: thesis: for x0 being Point of S
for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) )
let x0 be Point of S; :: thesis: for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) )

let r be Real; :: thesis: ( x0 in X & 0 < r implies ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) )

assume that
A3: x0 in X and
A4: 0 < r ; :: thesis: ex s being Real st
( 0 < s & ( for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) )

consider s being Real such that
A5: 0 < s and
A6: 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, A4;
take s = s; :: thesis: ( 0 < s & ( for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) )

thus 0 < s by A5; :: thesis: for x1 being Point of S st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r

let x1 be Point of S; :: thesis: ( x1 in X & ||.(x1 - x0).|| < s implies |.((f /. x1) - (f /. x0)).| < r )
assume ( x1 in X & ||.(x1 - x0).|| < s ) ; :: thesis: |.((f /. x1) - (f /. x0)).| < r
hence |.((f /. x1) - (f /. x0)).| < r by A3, A6; :: thesis: verum
end;
X c= dom f by A1;
hence f is_continuous_on X by ; :: thesis: verum