reserve n,m,k for Element of NAT;
reserve x, X,X1,Z,Z1 for set;
reserve s,g,r,p,x0,x1,x2 for Real;
reserve s1,s2,q1 for Real_Sequence;
reserve Y for Subset of REAL;
reserve f,f1,f2 for PartFunc of REAL,REAL;

theorem
  f|X is Lipschitzian & X1 c= X implies f|X1 is Lipschitzian
proof
  assume that
A1: f|X is Lipschitzian and
A2: X1 c= X;
  f|X1 = f|X|X1 by A2,RELAT_1:74;
  hence thesis by A1;
end;
