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

theorem Th16:
  f is_continuous_in x0 implies r(#)f is_continuous_in x0
proof
  assume
A1: f is_continuous_in x0;
  then x0 in dom f;
  hence
A2: x0 in dom (r(#)f) by VFUNCT_1:def 4;
  let s1;
  assume that
A3: rng s1 c= dom(r(#)f) and
A4: s1 is convergent & lim s1=x0;
A5: rng s1 c= dom f by A3,VFUNCT_1:def 4;
  then
A6: f/.x0 = lim (f/*s1) by A1,A4;
   reconsider rr=r as Real;
A7: f/*s1 is convergent by A1,A4,A5;
  then r*(f/*s1) is convergent by NORMSP_1:22;
  hence (r(#)f)/*s1 is convergent by A5,Th13;
  thus (r(#)f)/.x0 = r*f/.x0 by A2,VFUNCT_1:def 4
    .= lim (r*(f/*s1)) by A7,A6,NORMSP_1:28
    .= lim ((r(#)f)/*s1) by A5,Th13;
end;
