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_3:28;
  g1+g2 = f1+f2 by Th5;
  hence thesis by A2;
end;
