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
  X c= dom f1 & X1 c= dom f2 & f1|X is continuous & f2|X1 is continuous
  implies (f1+f2) | (X /\ X1) is continuous & (f1-f2) | (X /\ X1) is continuous
proof
  assume A1: X c= dom f1 & X1 c= dom f2 & f1|X is continuous & f2|X1
  is continuous;
   reconsider g1=f1,g2=f2 as PartFunc of REAL,REAL-NS n
   by REAL_NS1:def 4;
A2: g1|X is continuous by A1,Th23;
    g2|X1 is continuous by A1,Th23; then
A3: (g1+g2) | (X /\ X1) is continuous & (g1-g2) | (X /\ X1) is continuous
    by A1,A2,NFCONT_3:20;
A4: g1+g2 = f1+f2 by Th5;
    g1-g2 = f1-f2 by Th10;
    hence thesis by A3,A4,Th23;
end;
