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 Th26:
  for h be PartFunc of CNS1,CNS2, seq be sequence of CNS1, z be
  Complex st rng seq c= dom h holds (z(#)h)/*seq = z*(h/*seq)
proof
  let h be PartFunc of CNS1,CNS2;
  let seq be sequence of CNS1;
  let z be Complex;
  assume
A1: rng seq c= dom h;
  then
A2: rng seq c= dom (z(#)h) by VFUNCT_2:def 2;
  now
    let n be Nat;
A3:   n in NAT by ORDINAL1:def 12;
A4: seq.n in dom (z(#)h) by A2,Th4;
    thus ((z(#)h)/*seq).n = (z(#)h)/.(seq.n) by A2,FUNCT_2:109,A3
      .= z * (h/.(seq.n)) by A4,VFUNCT_2:def 2
      .= z * (h/*seq).n by A1,FUNCT_2:109,A3;
  end;
  hence thesis by CLVECT_1:def 14;
end;
