reserve M for non empty set;
reserve V for ComplexNormSpace;
reserve f,f1,f2,f3 for PartFunc of M,V;
reserve z,z1,z2 for Complex;

theorem
  z<>0c implies (z(#)f)"{0.V} = f"{0.V}
proof
  assume
A1: z<>0c;
  now
    let x be Element of M;
    thus x in (z(#)f)"{0.V} implies x in f"{0.V}
    proof
      assume
A2:   x in (z(#)f)"{0.V};
      then
A3:   x in dom (z(#)f) by PARTFUN2:26;
      (z(#)f)/.x in {0.V} by A2,PARTFUN2:26;
      then (z(#)f)/.x = 0.V by TARSKI:def 1;
      then z*f/.x = 0.V by A3,Def2;
      then z*f/.x = z*0.V by CLVECT_1:1;
      then f/.x = 0.V by A1,CLVECT_1:11;
      then
A4:   f/.x in {0.V} by TARSKI:def 1;
      x in dom f by A3,Def2;
      hence thesis by A4,PARTFUN2:26;
    end;
    assume
A5: x in (f)"{0.V};
    then x in dom f by PARTFUN2:26;
    then
A6: x in dom (z(#)f) by Def2;
    f/.x in {0.V} by A5,PARTFUN2:26;
    then z*f/.x = z*0.V by TARSKI:def 1;
    then z*f/.x = 0.V by CLVECT_1:1;
    then (z(#)f)/.x = 0.V by A6,Def2;
    then (z(#)f)/.x in {0.V} by TARSKI:def 1;
    hence x in (z(#)f)"{0.V} by A6,PARTFUN2:26;
  end;
  hence thesis by SUBSET_1:3;
end;
