reserve X, Y for RealNormSpace;

theorem Th9:
  for V be Subset of X,a be Real, V1 be Subset of
  LinearTopSpaceNorm X st V=V1 holds a*V=a*V1
proof
  let V be Subset of X,a be Real,
V1 be Subset of LinearTopSpaceNorm X such
  that
A1: V=V1;
  thus a*V c=a*V1
  proof
    let z be object;
    assume z in a*V;
    then consider v be Point of X such that
A2: z=a*v and
A3: v in V;
    reconsider v1=v as Point of LinearTopSpaceNorm X by A1,A3;
    a*v=a*v1 by NORMSP_2:def 4;
    hence thesis by A1,A2,A3;
  end;
  let z be object;
  assume z in a*V1;
  then consider v be Point of LinearTopSpaceNorm X such that
A4: z=a*v and
A5: v in V1;
  reconsider v1=v as Point of X by A1,A5;
  a*v=a*v1 by NORMSP_2:def 4;
  hence thesis by A1,A4,A5;
end;
