let X be set ; :: thesis: for RNS being RealNormSpace
for f being PartFunc of the carrier of RNS,COMPLEX st f is_uniformly_continuous_on X holds
f is_continuous_on X
let RNS be RealNormSpace; :: thesis: for f being PartFunc of the carrier of RNS,COMPLEX st f is_uniformly_continuous_on X holds
f is_continuous_on X
let f be PartFunc of the carrier of RNS,COMPLEX ; :: thesis: ( f is_uniformly_continuous_on X implies f is_continuous_on X )
assume A1: f is_uniformly_continuous_on X ; :: thesis: f is_continuous_on X
then A2: ( X c= dom f & ( for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1, x2 being Point of RNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
|.((f /. x1) - (f /. x2)).| < r ) ) ) ) by Def6;

now

let x0 be Point of RNS; :: thesis: for r being Real st x0 in X & 0 < r holds
ex s being Real st
( 0 < s & ( for x1 being Point of RNS 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 RNS st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) ) )
assume A3: ( x0 in X & 0 < r ) ; :: thesis: ex s being Real st
( 0 < s & ( for x1 being Point of RNS st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) )
then consider s being Real such that
A4: ( 0 < s & ( for x1, x2 being Point of RNS st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
|.((f /. x1) - (f /. x2)).| < r ) ) by A1, Def6;
take s = s; :: thesis: ( 0 < s & ( for x1 being Point of RNS st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r ) )
thus 0 < s by A4; :: thesis: for x1 being Point of RNS st x1 in X & ||.(x1 - x0).|| < s holds
|.((f /. x1) - (f /. x0)).| < r
let x1 be Point of RNS; :: 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, A4; :: thesis: verum

end;

hence f is_continuous_on X by A2, NCFCONT1:70; :: thesis: verum