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 Th38:
  f is_Lipschitzian_on X & X1 c= X implies f is_Lipschitzian_on X1
proof
  assume that
A1: f is_Lipschitzian_on X and
A2: X1 c= X;
  X c= dom f by A1;
  hence X1 c= dom f by A2;
  consider s such that
A3: 0<s and
A4: for x1,x2 st x1 in X & x2 in X holds ||. f/.x1-f/.x2.||<=s*||. x1-x2
  .|| by A1;
  take s;
  thus 0<s by A3;
  let x1,x2;
  assume x1 in X1 & x2 in X1;
  hence thesis by A2,A4;
end;
