let X, X1 be set ; 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; 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; ( f is_uniformly_continuous_on X & X1 c= X implies f is_uniformly_continuous_on X1 )
assume that
A1:
f is_uniformly_continuous_on X
and
A2:
X1 c= X
; f is_uniformly_continuous_on X1
X c= dom f
by A1;
hence
X1 c= dom f
by A2, XBOOLE_1:1; NCFCONT2:def 1 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; ( 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
; 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
A3:
0 < s
and
A4:
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;
take
s
; ( 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 A3; 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; ( 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 )
; ||.((f /. x1) - (f /. x2)).|| < r
hence
||.((f /. x1) - (f /. x2)).|| < r
by A2, A4; verum