reserve a for Real;
reserve p,q for Point of TOP-REAL 2;

theorem Th13:
  for n being Element of NAT,K0 being Subset of TOP-REAL n, f
being Function of (TOP-REAL n)|K0,R^1 st (for p being Point of (TOP-REAL n)|K0
  holds f.p=(n NormF).p) holds f is continuous
proof
  let n be Element of NAT,K0 be Subset of TOP-REAL n, f be Function of (
  TOP-REAL n)|K0,R^1;
A1: (the carrier of TOP-REAL n)/\K0=K0 by XBOOLE_1:28;
  reconsider g=(n NormF) as Function of TOP-REAL n,R^1;
  assume for p being Point of (TOP-REAL n)|K0 holds f.p=(n NormF).p;
  then
A2: for x being object st x in dom f holds f.x=(n NormF).x;
  dom f= the carrier of (TOP-REAL n)|K0 & the carrier of (TOP-REAL n)|K0=
  K0 by FUNCT_2:def 1,PRE_TOPC:8;
  then dom f=dom (n NormF) /\ K0 by A1,FUNCT_2:def 1;
  then f=g|K0 by A2,FUNCT_1:46;
  hence thesis by TOPMETR:7;
end;
