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 by Def1;
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
abs ((||.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
abs ((||.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
abs ((||.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, Def1;
take s ; :: thesis: ( 0 < s & ( for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
abs ((||.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
abs ((||.f.|| /. x1) - (||.f.|| /. x2)) < r
let x1, x2 be Point of S; :: thesis: ( x1 in X & x2 in X & ||.(x1 - x2).|| < s implies abs ((||.f.|| /. x1) - (||.f.|| /. x2)) < r )
assume that
A5: x1 in X and
A6: x2 in X and
A7: ||.(x1 - x2).|| < s ; :: thesis: abs ((||.f.|| /. x1) - (||.f.|| /. x2)) < r
abs ((||.f.|| /. x1) - (||.f.|| /. x2)) = abs ((||.f.|| . x1) - (||.f.|| /. x2)) by A2, A5, PARTFUN1:def 8
.= abs ((||.f.|| . x1) - (||.f.|| . x2)) by A2, A6, PARTFUN1:def 8
.= abs (||.(f /. x1).|| - (||.f.|| . x2)) by A2, A5, NORMSP_0:def 3
.= abs (||.(f /. x1).|| - ||.(f /. x2).||) by A2, A6, NORMSP_0:def 3 ;
then A8: abs ((||.f.|| /. x1) - (||.f.|| /. x2)) <= ||.((f /. x1) - (f /. x2)).|| by NORMSP_1:13;
||.((f /. x1) - (f /. x2)).|| < r by A4, A5, A6, A7;
hence abs ((||.f.|| /. x1) - (||.f.|| /. x2)) < r by A8, XXREAL_0:2; :: thesis: verum