reserve n,m,k for Nat;
reserve x,X,X1 for set;
reserve r,p for Real;
reserve s,g,x0,x1,x2 for Real;
reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of REAL,the carrier of S;
reserve s1,s2 for Real_Sequence;
reserve Y for Subset of REAL;

theorem
for X,X1,f1,f2 st X c= dom f1 & X1 c= dom f2
 & f1|X is continuous & f2|X1 is continuous
   holds (f1+f2)|(X /\ X1) is continuous & (f1-f2)|(X /\ X1) is continuous
proof
   let X,X1,f1,f2;
   assume X c= dom f1 & X1 c= dom f2; then
A1:X /\ X1 c= dom f1 /\ dom f2 by XBOOLE_1:27;
   assume f1|X is continuous & f2|X1 is continuous; then
   f1|(X /\ X1) is continuous & f2|(X /\ X1) is continuous by Th18,XBOOLE_1:17;
   hence thesis by A1,Th19;
end;
