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

theorem
  f is_Lipschitzian_on X implies -f is_Lipschitzian_on X & ||.f.||
  is_Lipschitzian_on X
proof
  assume
A1: f is_Lipschitzian_on X;
  then consider s such that
A2: 0<s and
A3: for x1,x2 st x1 in X & x2 in X holds ||. f/.x1-f/.x2.||<=s*||. x1-x2
  .||;
  -f=(-1)(#)f by VFUNCT_1:23;
  hence -f is_Lipschitzian_on X by A1,Th41;
  X c= dom f by A1;
  hence
A4: X c= dom (||.f.|| ) by NORMSP_0:def 3;
  take s;
  thus 0<s by A2;
  let x1,x2;
  assume that
A5: x1 in X and
A6: x2 in X;
  |. (||.f.|| )/.x1-(||.f.|| )/.x2 .| =|. (||.f.|| ).x1-(||.f.|| )/.x2
  .| by A4,A5,PARTFUN1:def 6
    .=|. (||.f.|| ).x1-(||.f.|| ).x2 .| by A4,A6,PARTFUN1:def 6
    .= |. ||. f/.x1.||-(||.f.|| ).x2 .| by A4,A5,NORMSP_0:def 3
    .= |.||. f/.x1.|| - ||. f/.x2 .||.| by A4,A6,NORMSP_0:def 3;
  then
A7: |. (||.f.|| )/.x1-(||.f.|| )/.x2 .| <= ||. f/.x1-f/.x2.|| by NORMSP_1:9;
  ||. f/.x1-f/.x2 .||<=s*||. x1-x2.|| by A3,A5,A6;
  hence thesis by A7,XXREAL_0:2;
end;
