let CNS be ComplexNormSpace; :: thesis: for RNS being RealNormSpace
for X being set
for f being PartFunc of RNS,CNS st X c= dom f & f | X is constant holds
f is_Lipschitzian_on X
let RNS be RealNormSpace; :: thesis: for X being set
for f being PartFunc of RNS,CNS st X c= dom f & f | X is constant holds
f is_Lipschitzian_on X
let X be set ; :: thesis: for f being PartFunc of RNS,CNS st X c= dom f & f | X is constant holds
f is_Lipschitzian_on X
let f be PartFunc of RNS,CNS; :: thesis: ( X c= dom f & f | X is constant implies f is_Lipschitzian_on X )
assume A1: ( X c= dom f & f | X is constant ) ; :: thesis: f is_Lipschitzian_on X

now

let x1, x2 be Point of RNS; :: thesis: ( x1 in X & x2 in X implies ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).|| )
assume A2: ( x1 in X & x2 in X ) ; :: thesis: ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).||
then A3: ( x1 in X /\ (dom f) & x2 in X /\ (dom f) ) by A1, XBOOLE_0:def 4;
f /. x1 = f . x1 by A1, A2, PARTFUN1:def 8
.= f . x2 by A1, A3, PARTFUN2:77
.= f /. x2 by A1, A2, PARTFUN1:def 8 ;
then ||.((f /. x1) - (f /. x2)).|| = ||.(0. CNS).|| by RLVECT_1:28
.= 0 by CLVECT_1:103 ;
hence ||.((f /. x1) - (f /. x2)).|| <= 1 * ||.(x1 - x2).|| by NORMSP_1:8; :: thesis: verum

end;

hence f is_Lipschitzian_on X by A1, Def29; :: thesis: verum