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 CNS,REAL st X c= dom f & (for x0
be Point of CNS st x0 in X holds f/.x0 = ||. x0.||) holds f is_continuous_on X
proof
  let f be PartFunc of the carrier of CNS,REAL;
  assume that
A1: X c= dom f and
A2: for x0 be Point of CNS st x0 in X holds f/.x0 = ||. x0 .||;
  X=dom f /\ X by A1,XBOOLE_1:28;
  then
A3: X=dom(f|X) by RELAT_1:61;
  now
    let x0 be Point of CNS;
    assume
A4: x0 in dom(f|X);
    hence (f|X)/.x0=f/.x0 by PARTFUN2:15
      .=||. x0 .|| by A2,A3,A4;
  end;
  then f|X is_continuous_on X by A3,Th132;
  hence thesis by Th54;
end;
