reserve n,m,k for Nat;
reserve x,X,X1 for set;
reserve r,p for Real;
reserve s,g,x0,x1,x2 for Real;
reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of REAL,the carrier of S;
reserve s1,s2 for Real_Sequence;
reserve Y for Subset of REAL;

theorem
f|X is Lipschitzian
 implies -(f|X) is Lipschitzian & (-f)|X is Lipschitzian
   & (||.f.||)|X is Lipschitzian
proof
   assume A1: f|X is Lipschitzian;
A2:-(f|X) = (-1)(#)(f|X) by VFUNCT_1:23;
   hence -(f|X) is Lipschitzian by A1;
   thus (-f)|X is Lipschitzian by A1,A2,VFUNCT_1:29;
   (||.f.||)|X = ||. f|X .|| by VFUNCT_1:29;
   hence thesis by A1;
end;
