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
  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 & (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 Th16,
XBOOLE_1:17;
  hence thesis by A1,Th18;
end;
