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
  for S be RealNormSpace,
      z be Point of REAL-NS n,
      f1 be PartFunc of REAL,REAL n,
      f2 be PartFunc of REAL-NS n,S
     st x0 in dom (f2*f1) & f1 is_continuous_in x0
        & z=f1/.x0 & f2 is_continuous_in z
    holds f2*f1 is_continuous_in x0
proof
let S be RealNormSpace,
    z be Point of REAL-NS n,
    f1 be PartFunc of REAL,REAL n,
    f2 be PartFunc of REAL-NS n,S;
  assume A1: x0 in dom (f2*f1) & f1 is_continuous_in x0
        & z=f1/.x0 & f2 is_continuous_in z;
   reconsider g1= f1 as PartFunc of REAL,REAL-NS n
   by REAL_NS1:def 4;
  f1/.x0 = g1/.x0 by REAL_NS1:def 4;
  hence thesis by A1,NFCONT_3:15;
end;
