let X be set ; :: thesis: for CNS1, CNS2 being ComplexNormSpace
for f being PartFunc of CNS1,CNS2 st f is_Lipschitzian_on X holds
f is_uniformly_continuous_on X
let CNS1, CNS2 be ComplexNormSpace; :: thesis: for f being PartFunc of CNS1,CNS2 st f is_Lipschitzian_on X holds
f is_uniformly_continuous_on X
let f be PartFunc of CNS1,CNS2; :: 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 NCFCONT1:def 17; :: according to NCFCONT2:def 1 :: thesis: for r being Real st 0 < r holds
ex s being Real st
( 0 < s & ( for x1, x2 being Point of CNS1 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 CNS1 st x1 in X & x2 in X holds
||.((f /. x1) - (f /. x2)).|| <= r * ||.(x1 - x2).|| by A1, NCFCONT1:def 17;
let p be Real; :: thesis: ( 0 < p implies ex s being Real st
( 0 < s & ( for x1, x2 being Point of CNS1 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 CNS1 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 CNS1 st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.((f /. x1) - (f /. x2)).|| < p ) )
thus 0 < s by A2, A4; :: thesis: for x1, x2 being Point of CNS1 st x1 in X & x2 in X & ||.(x1 - x2).|| < s holds
||.((f /. x1) - (f /. x2)).|| < p
let x1, x2 be Point of CNS1; :: 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