let X, X1 be set ; :: thesis: for CNS1, CNS2 being ComplexNormSpace
for f being PartFunc of CNS1,CNS2 st f is_uniformly_continuous_on X & X1 c= X holds
f is_uniformly_continuous_on X1
let CNS1, CNS2 be ComplexNormSpace; :: thesis: for f being PartFunc of CNS1,CNS2 st f is_uniformly_continuous_on X & X1 c= X holds
f is_uniformly_continuous_on X1
let f be PartFunc of CNS1,CNS2; :: thesis: ( f is_uniformly_continuous_on X & X1 c= X implies f is_uniformly_continuous_on X1 )
assume A1: ( f is_uniformly_continuous_on X & X1 c= X ) ; :: thesis: f is_uniformly_continuous_on X1
then X c= dom f by Def1;
hence X1 c= dom f by A1, XBOOLE_1:1; :: 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 X1 & x2 in X1 & ||.(x1 - x2).|| < s holds
||.((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 CNS1 st x1 in X1 & x2 in X1 & ||.(x1 - x2).|| < s holds
||.((f /. x1) - (f /. x2)).|| < r ) ) )
assume 0 < r ; :: thesis: ex s being Real st
( 0 < s & ( for x1, x2 being Point of CNS1 st x1 in X1 & x2 in X1 & ||.(x1 - x2).|| < s holds
||.((f /. x1) - (f /. x2)).|| < r ) )
then consider s being Real such that
A2: ( 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 ) ) by A1, Def1;
take s ; :: thesis: ( 0 < s & ( for x1, x2 being Point of CNS1 st x1 in X1 & x2 in X1 & ||.(x1 - x2).|| < s holds
||.((f /. x1) - (f /. x2)).|| < r ) )
thus 0 < s by A2; :: thesis: for x1, x2 being Point of CNS1 st x1 in X1 & x2 in X1 & ||.(x1 - x2).|| < s holds
||.((f /. x1) - (f /. x2)).|| < r
let x1, x2 be Point of CNS1; :: thesis: ( x1 in X1 & x2 in X1 & ||.(x1 - x2).|| < s implies ||.((f /. x1) - (f /. x2)).|| < r )
assume ( x1 in X1 & x2 in X1 & ||.(x1 - x2).|| < s ) ; :: thesis: ||.((f /. x1) - (f /. x2)).|| < r
hence ||.((f /. x1) - (f /. x2)).|| < r by A1, A2; :: thesis: verum