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 iff ex r be Real
     st 0<r & for x1,x2 st x1 in dom(f|X) &
   x2 in dom(f|X) holds |. f/.x1-f/.x2 .| <= r*|.x1-x2.|
proof
  hereby assume f|X is Lipschitzian; then
  consider r be Real such that
A1: 0<r & for x1,x2 st x1 in dom (f|X) &
    x2 in dom (f|X) holds |.(f|X)/.x1-(f|X)/.x2.|<=r*|.x1-x2.| by Th33;
  take r;
  thus 0<r by A1;
  thus for x1,x2 st x1 in dom(f|X) &
        x2 in dom(f|X) holds |. f/.x1-f/.x2 .|<=r*|.x1-x2.|
  proof
    let x1,x2;
    assume A2: x1 in dom(f|X) & x2 in dom(f|X); then
    |.(f|X)/.x1-(f|X)/.x2.|<=r*|.x1-x2.| by A1; then
    |.f/.x1-(f|X)/.x2.|<=r*|.x1-x2.| by A2,PARTFUN2:15;
    hence |.f/.x1-f/.x2.|<=r*|.x1-x2.| by A2,PARTFUN2:15;
  end;
  end;
  given r be Real such that
A3: 0<r & for x1,x2 st x1 in dom(f|X) &
    x2 in dom(f|X) holds |. f/.x1-f/.x2 .|<=r*|.x1-x2.|;
  now
   let x1,x2;
   assume A4: x1 in dom(f|X) & x2 in dom(f|X);
   then |. f/.x1-f/.x2 .|<=r*|.x1-x2.| by A3;
   then |.(f|X)/.x1-f/.x2.|<=r*|.x1-x2.| by A4,PARTFUN2:15;
   hence |.(f|X)/.x1-(f|X)/.x2.|<=r*|.x1-x2.| by A4,PARTFUN2:15;
 end;
 hence f|X is Lipschitzian by A3,Th33;
end;
