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
  x0 in dom f1 /\ dom f2 & f1 is_continuous_in x0 & f2 is_continuous_in x0
  implies f1-f2 is_continuous_in x0
proof
  assume A1: x0 in dom f1 /\ dom f2 & f1 is_continuous_in x0 &
    f2 is_continuous_in x0;
   reconsider g1=f1,g2=f2 as PartFunc of REAL,REAL-NS n
     by REAL_NS1:def 4;
A2:(g1+g2) is_continuous_in x0 & (g1-g2) is_continuous_in x0
     by A1,NFCONT_3:12;
   g1-g2 = f1-f2 by Th10;
   hence thesis by A2;
end;
