let CNS be ComplexNormSpace; for RNS being RealNormSpace
for X, X1 being set
for f being PartFunc of RNS,CNS st f is_Lipschitzian_on X & X1 c= X holds
f is_Lipschitzian_on X1
let RNS be RealNormSpace; for X, X1 being set
for f being PartFunc of RNS,CNS st f is_Lipschitzian_on X & X1 c= X holds
f is_Lipschitzian_on X1
let X, X1 be set ; for f being PartFunc of RNS,CNS st f is_Lipschitzian_on X & X1 c= X holds
f is_Lipschitzian_on X1
let f be PartFunc of RNS,CNS; ( f is_Lipschitzian_on X & X1 c= X implies f is_Lipschitzian_on X1 )
assume that
A1:
f is_Lipschitzian_on X
and
A2:
X1 c= X
; f is_Lipschitzian_on X1
X c= dom f
by A1;
hence
X1 c= dom f
by A2; NCFCONT1:def 19 ex r being Real st
( 0 < r & ( for x1, x2 being Point of RNS st x1 in X1 & x2 in X1 holds
||.((f /. x1) - (f /. x2)).|| <= r * ||.(x1 - x2).|| ) )
consider s being Real such that
A3:
0 < s
and
A4:
for x1, x2 being Point of RNS st x1 in X & x2 in X holds
||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).||
by A1;
take
s
; ( 0 < s & ( for x1, x2 being Point of RNS st x1 in X1 & x2 in X1 holds
||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| ) )
thus
0 < s
by A3; for x1, x2 being Point of RNS st x1 in X1 & x2 in X1 holds
||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).||
let x1, x2 be Point of RNS; ( x1 in X1 & x2 in X1 implies ||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| )
assume
( x1 in X1 & x2 in X1 )
; ||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).||
hence
||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).||
by A2, A4; verum