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 f & f|X is continuous
  implies |. f .| |X is continuous & (-f) | X is continuous
proof
   assume A1: X c= dom f & f|X is continuous;
   reconsider g= f as PartFunc of REAL,REAL-NS n
   by REAL_NS1:def 4;
   g|X is continuous by A1,Th23; then
A2: ||. g .|| |X is continuous & (-g) | X is continuous by A1,NFCONT_3:22;
   hence |. f .| |X is continuous by Th9;
   -g = - f by Th8;
   hence (-f) | X is continuous by A2,Th23;
end;
