reserve n,m,k for Nat;
reserve x,X,X1 for set;
reserve r,p for Real;
reserve s,g,x0,x1,x2 for Real;
reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of REAL,the carrier of S;
reserve s1,s2 for Real_Sequence;
reserve Y for Subset of REAL;

theorem Th18:
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;
