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;
reserve X,X1 for set;

theorem Th69:
  for f be PartFunc of CNS,RNS st f is_continuous_on X holds r(#)f
  is_continuous_on X
proof
  let f be PartFunc of CNS,RNS;
  assume
A1: f is_continuous_on X;
  then
A2: X c= dom f;
  then
A3: X c= dom(r(#)f) by VFUNCT_1:def 4;
  now
    let s1 be sequence of CNS;
    assume that
A4: rng s1 c= X and
A5: s1 is convergent and
A6: lim s1 in X;
A7: f/*s1 is convergent by A1,A4,A5,A6,Th42;
    then
A8: r*(f/*s1) is convergent by NORMSP_1:22;
    f/.(lim s1) = lim (f/*s1) by A1,A4,A5,A6,Th42;
    then (r(#)f)/.(lim s1) = r * lim (f/*s1) by A3,A6,VFUNCT_1:def 4
      .= lim (r*(f/*s1)) by A7,NORMSP_1:28
      .= lim ((r(#)f)/*s1) by A2,A4,Th27,XBOOLE_1:1;
    hence
    (r(#)f)/*s1 is convergent & (r(#)f)/.(lim s1)=lim((r(#)f)/*s1) by A2,A4,A8
,Th27,XBOOLE_1:1;
  end;
  hence thesis by A3,Th42;
end;
