reserve n,m for Element of NAT;
reserve r,s for Real;
reserve z for Complex;
reserve CNS,CNS1,CNS2 for ComplexNormSpace;
reserve RNS for RealNormSpace;
reserve X,X1 for set;

theorem
  for f be PartFunc of the carrier of RNS,COMPLEX holds (f
  is_continuous_on X iff f|X is_continuous_on X)
proof
  let f be PartFunc of the carrier of RNS,COMPLEX;
  thus f is_continuous_on X implies f|X is_continuous_on X
  proof
    assume
A1: f is_continuous_on X;
    then X c= dom f;
    then X c= dom f /\ X by XBOOLE_1:28;
    hence X c= dom (f|X) by RELAT_1:61;
    let r be Point of RNS;
    assume r in X;
    then f|X is_continuous_in r by A1;
    hence thesis by RELAT_1:72;
  end;
  assume
A2: f|X is_continuous_on X;
  then X c= dom(f|X);
  then dom f /\ X c= dom f & X c= dom f /\ X by RELAT_1:61,XBOOLE_1:17;
  hence X c= dom f;
  let r be Point of RNS;
  assume r in X;
  then (f|X)|X is_continuous_in r by A2;
  hence thesis by RELAT_1:72;
end;
