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
  f1|X is Lipschitzian & f2|X1 is Lipschitzian implies
  (f1-f2) | (X /\ X1) is Lipschitzian
proof
  assume
  A1:f1|X is Lipschitzian & f2|X1 is Lipschitzian;
    reconsider g1=f1,g2=f2 as PartFunc of REAL,REAL-NS n
    by REAL_NS1:def 4;
 g1|X is Lipschitzian & g2|X1 is Lipschitzian by A1;
   then
 A2: (g1-g2) | (X /\ X1) is Lipschitzian by NFCONT_3:29;
 g1-g2 = f1-f2 by Th10;
 hence thesis by A2;
end;
