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
  f|X is continuous & X1 c= X implies f|X1 is continuous
proof
  assume
A1: f|X is continuous;
  assume X1 c= X;
  then f|X1 = f|X|X1 by RELAT_1:74;
  hence thesis by A1;
end;
