let CNS be ComplexNormSpace; :: thesis: for RNS being RealNormSpace
for X being set
for f being PartFunc of RNS,CNS st f is_Lipschitzian_on X holds
( - f is_Lipschitzian_on X & ||.f.|| is_Lipschitzian_on X )
let RNS be RealNormSpace; :: thesis: for X being set
for f being PartFunc of RNS,CNS st f is_Lipschitzian_on X holds
( - f is_Lipschitzian_on X & ||.f.|| is_Lipschitzian_on X )
let X be set ; :: thesis: for f being PartFunc of RNS,CNS st f is_Lipschitzian_on X holds
( - f is_Lipschitzian_on X & ||.f.|| is_Lipschitzian_on X )
let f be PartFunc of RNS,CNS; :: thesis: ( f is_Lipschitzian_on X implies ( - f is_Lipschitzian_on X & ||.f.|| is_Lipschitzian_on X ) )
assume A1: f is_Lipschitzian_on X ; :: thesis: ( - f is_Lipschitzian_on X & ||.f.|| is_Lipschitzian_on X )
- f = (- 1r ) (#) f by VFUNCT_2:23;
hence - f is_Lipschitzian_on X by A1, Th129; :: thesis: ||.f.|| is_Lipschitzian_on X
X c= dom f by A1, Def29;
hence A2: X c= dom ||.f.|| by Def4; :: according to NFCONT_1:def 14 :: thesis: ex b₁ being Element of REAL st
( not b₁ <= 0 & ( for b₂, b₃ being Element of the carrier of RNS holds
( not b₂ in X or not b₃ in X or abs ((||.f.|| /. b₂) - (||.f.|| /. b₃)) <= b₁ * ||.(b₂ - b₃).|| ) ) )
consider s being Real such that
A3: ( 0 < s & ( for x1, x2 being Point of RNS st x1 in X & x2 in X holds
||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| ) ) by A1, Def29;
take s ; :: thesis: ( not s <= 0 & ( for b₁, b₂ being Element of the carrier of RNS holds
( not b₁ in X or not b₂ in X or abs ((||.f.|| /. b₁) - (||.f.|| /. b₂)) <= s * ||.(b₁ - b₂).|| ) ) )
thus 0 < s by A3; :: thesis: for b₁, b₂ being Element of the carrier of RNS holds
( not b₁ in X or not b₂ in X or abs ((||.f.|| /. b₁) - (||.f.|| /. b₂)) <= s * ||.(b₁ - b₂).|| )
let x1, x2 be Point of RNS; :: thesis: ( not x1 in X or not x2 in X or abs ((||.f.|| /. x1) - (||.f.|| /. x2)) <= s * ||.(x1 - x2).|| )
assume A4: ( x1 in X & x2 in X ) ; :: thesis: abs ((||.f.|| /. x1) - (||.f.|| /. x2)) <= s * ||.(x1 - x2).||
then abs ((||.f.|| /. x1) - (||.f.|| /. x2)) = abs ((||.f.|| . x1) - (||.f.|| /. x2)) by A2, PARTFUN1:def 8
.= abs ((||.f.|| . x1) - (||.f.|| . x2)) by A2, A4, PARTFUN1:def 8
.= abs (||.(f /. x1).|| - (||.f.|| . x2)) by A2, A4, Def4
.= abs (||.(f /. x1).|| - ||.(f /. x2).||) by A2, A4, Def4 ;
then A5: abs ((||.f.|| /. x1) - (||.f.|| /. x2)) <= ||.((f /. x1) - (f /. x2)).|| by CLVECT_1:111;
||.((f /. x1) - (f /. x2)).|| <= s * ||.(x1 - x2).|| by A3, A4;
hence abs ((||.f.|| /. x1) - (||.f.|| /. x2)) <= s * ||.(x1 - x2).|| by A5, XXREAL_0:2; :: thesis: verum