let X be set ; :: thesis: for S, T being RealNormSpace

for f being PartFunc of S,T st f is_Lipschitzian_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_Lipschitzian_on X holds

f is_uniformly_continuous_on X

let f be PartFunc of S,T; :: thesis: ( f is_Lipschitzian_on X implies f is_uniformly_continuous_on X )

assume A1: f is_Lipschitzian_on X ; :: thesis: f is_uniformly_continuous_on X

hence X c= dom f by NFCONT_1:def 9; :: 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 X & x2 in X & ||.(x1 - x2).|| < s holds

||.((f /. x1) - (f /. x2)).|| < r ) )

consider r being Real such that

A2: 0 < r and

A3: for x1, x2 being Point of S st x1 in X & x2 in X holds

||.((f /. x1) - (f /. x2)).|| <= r * ||.(x1 - x2).|| by A1, NFCONT_1:def 9;

let p be Real; :: thesis: ( 0 < p 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

||.((f /. x1) - (f /. x2)).|| < p ) ) )

assume A4: 0 < p ; :: 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

||.((f /. x1) - (f /. x2)).|| < p ) )

take s = p / r; :: thesis: ( 0 < s & ( for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds

||.((f /. x1) - (f /. x2)).|| < p ) )

thus 0 < s by A2, A4, XREAL_1:139; :: thesis: for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds

||.((f /. x1) - (f /. x2)).|| < p

let x1, x2 be Point of S; :: thesis: ( x1 in X & x2 in X & ||.(x1 - x2).|| < s implies ||.((f /. x1) - (f /. x2)).|| < p )

assume ( x1 in X & x2 in X & ||.(x1 - x2).|| < s ) ; :: thesis: ||.((f /. x1) - (f /. x2)).|| < p

then ( r * ||.(x1 - x2).|| < s * r & ||.((f /. x1) - (f /. x2)).|| <= r * ||.(x1 - x2).|| ) by A2, A3, XREAL_1:68;

then ||.((f /. x1) - (f /. x2)).|| < (p / r) * r by XXREAL_0:2;

hence ||.((f /. x1) - (f /. x2)).|| < p by A2, XCMPLX_1:87; :: thesis: verum

for f being PartFunc of S,T st f is_Lipschitzian_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_Lipschitzian_on X holds

f is_uniformly_continuous_on X

let f be PartFunc of S,T; :: thesis: ( f is_Lipschitzian_on X implies f is_uniformly_continuous_on X )

assume A1: f is_Lipschitzian_on X ; :: thesis: f is_uniformly_continuous_on X

hence X c= dom f by NFCONT_1:def 9; :: 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 X & x2 in X & ||.(x1 - x2).|| < s holds

||.((f /. x1) - (f /. x2)).|| < r ) )

consider r being Real such that

A2: 0 < r and

A3: for x1, x2 being Point of S st x1 in X & x2 in X holds

||.((f /. x1) - (f /. x2)).|| <= r * ||.(x1 - x2).|| by A1, NFCONT_1:def 9;

let p be Real; :: thesis: ( 0 < p 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

||.((f /. x1) - (f /. x2)).|| < p ) ) )

assume A4: 0 < p ; :: 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

||.((f /. x1) - (f /. x2)).|| < p ) )

take s = p / r; :: thesis: ( 0 < s & ( for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds

||.((f /. x1) - (f /. x2)).|| < p ) )

thus 0 < s by A2, A4, XREAL_1:139; :: thesis: for x1, x2 being Point of S st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds

||.((f /. x1) - (f /. x2)).|| < p

let x1, x2 be Point of S; :: thesis: ( x1 in X & x2 in X & ||.(x1 - x2).|| < s implies ||.((f /. x1) - (f /. x2)).|| < p )

assume ( x1 in X & x2 in X & ||.(x1 - x2).|| < s ) ; :: thesis: ||.((f /. x1) - (f /. x2)).|| < p

then ( r * ||.(x1 - x2).|| < s * r & ||.((f /. x1) - (f /. x2)).|| <= r * ||.(x1 - x2).|| ) by A2, A3, XREAL_1:68;

then ||.((f /. x1) - (f /. x2)).|| < (p / r) * r by XXREAL_0:2;

hence ||.((f /. x1) - (f /. x2)).|| < p by A2, XCMPLX_1:87; :: thesis: verum