reserve n,m,i,k for Element of NAT;
reserve x,X,X1 for set;
reserve r,p for Real;
reserve s,x0,x1,x2 for Real;
reserve f,f1,f2 for PartFunc of REAL,REAL n;
reserve h for PartFunc of REAL,REAL-NS n;
reserve W for non empty set;

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;
   reconsider g= f as PartFunc of REAL,REAL-NS n
     by REAL_NS1:def 4;
   g|X is Lipschitzian by A1; then
A2: -(g|X) is Lipschitzian & (||. g .||) |X is Lipschitzian &
   (-g) | X is Lipschitzian by NFCONT_3:31;
  -(g|X) = -(f|X) by Th8;
  hence -(f|X) is Lipschitzian by A2;
  thus (|. f .|) |X is Lipschitzian by A2,Th9;
  -g = - f by Th8;
  hence thesis by A2;
end;
