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
  f1|X is Lipschitzian & f2|X1 is Lipschitzian implies (f1+f2)|(X /\ X1)
  is Lipschitzian
proof
A1: f1|(X /\ X1) = f1|X|(X /\ X1) & f2|(X /\ X1) = f2|X1|(X /\ X1) by
RELAT_1:74,XBOOLE_1:17;
A2: (f1+f2)|(X /\ X1) = f1|(X /\ X1)+f2|(X /\ X1) by RFUNCT_1:44;
  assume f1|X is Lipschitzian & f2|X1 is Lipschitzian;
  hence thesis by A1,A2;
end;
