reserve n,n1,m,m1,k for Nat;
reserve x,X,X1 for set;
reserve g,g1,g2,t,x0,x1,x2 for Complex;
reserve s1,s2,q1,seq,seq1,seq2,seq3 for Complex_Sequence;
reserve Y for Subset of COMPLEX;
reserve f,f1,f2,h,h1,h2 for PartFunc of COMPLEX,COMPLEX;
reserve p,r,s for Real;
reserve Ns,Nseq for increasing sequence of NAT;

theorem
  for X,X1,f1,f2 st f1 is_continuous_on X & f2 is_continuous_on X1 holds
  f1+f2 is_continuous_on X /\ X1 & f1-f2 is_continuous_on X /\ X1 & f1(#)f2
  is_continuous_on X /\ X1
proof
  let X,X1,f1,f2;
  assume f1 is_continuous_on X & f2 is_continuous_on X1;
  then f1 is_continuous_on X /\ X1 & f2 is_continuous_on X /\ X1 by Th41,
XBOOLE_1:17;
  hence thesis by Th43;
end;
