let CNS be ComplexNormSpace; :: thesis: for RNS being RealNormSpace
for X, X1 being set
for f being PartFunc of CNS,RNS st f is_Lipschitzian_on X & X1 c= X holds
f is_Lipschitzian_on X1
let RNS be RealNormSpace; :: thesis: for X, X1 being set
for f being PartFunc of CNS,RNS st f is_Lipschitzian_on X & X1 c= X holds
f is_Lipschitzian_on X1
let X, X1 be set ; :: thesis: for f being PartFunc of CNS,RNS st f is_Lipschitzian_on X & X1 c= X holds
f is_Lipschitzian_on X1
let f be PartFunc of CNS,RNS; :: thesis: ( 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 ; :: thesis: f is_Lipschitzian_on X1
X c= dom f by A1;
hence X1 c= dom f by A2; :: according to NCFCONT1:def 18 :: thesis: ex r being Real st
( 0 < r & ( for x1, x2 being Point of CNS 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 CNS st x1 in X & x2 in X holds
||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| by A1;
take s ; :: thesis: ( 0 < s & ( for x1, x2 being Point of CNS st x1 in X1 & x2 in X1 holds
||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| ) )
thus 0 < s by A3; :: thesis: for x1, x2 being Point of CNS st x1 in X1 & x2 in X1 holds
||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).||
let x1, x2 be Point of CNS; :: thesis: ( x1 in X1 & x2 in X1 implies ||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| )
assume ( x1 in X1 & x2 in X1 ) ; :: thesis: ||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).||
hence ||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| by A2, A4; :: thesis: verum