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;

theorem Th36:
  for f be PartFunc of CNS,RNS, x0 be Point of CNS, r be Real st f
  is_continuous_in x0 holds r(#)f is_continuous_in x0
proof
  let f be PartFunc of CNS,RNS;
  let x0 be Point of CNS;
  let r be Real;
  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 be sequence of CNS;
  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;
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,Th27;
  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,Th27;
end;
