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 Th37:
  for f be PartFunc of RNS,CNS, x0 be Point of RNS, z be Complex
  st f is_continuous_in x0 holds z(#)f is_continuous_in x0
proof
  let f be PartFunc of RNS,CNS;
  let x0 be Point of RNS;
  let z be Complex;
  assume
A1: f is_continuous_in x0;
  then x0 in dom f;
  hence
A2: x0 in dom (z(#)f) by VFUNCT_2:def 2;
  let s1 be sequence of RNS;
  assume that
A3: rng s1 c= dom(z(#)f) and
A4: s1 is convergent & lim s1=x0;
A5: rng s1 c= dom f by A3,VFUNCT_2:def 2;
  then
A6: f/.x0 = lim (f/*s1) by A1,A4;
A7: f/*s1 is convergent by A1,A4,A5;
  then z*(f/*s1) is convergent by CLVECT_1:116;
  hence (z(#)f)/*s1 is convergent by A5,Th28;
  thus (z(#)f)/.x0 = z*f/.x0 by A2,VFUNCT_2:def 2
    .= lim (z*(f/*s1)) by A7,A6,CLVECT_1:122
    .= lim ((z(#)f)/*s1) by A5,Th28;
end;
