reserve r,s,t,u for Real;

theorem
  for X being LinearTopSpace,A being Subset of X st X is T_1 holds 0*Cl(
  A) = Cl(0*A)
proof
  let X be LinearTopSpace, A be Subset of X such that
A1: X is T_1;
  per cases;
  suppose
A2: A is empty;
    then
A3: Cl A = {}X by PRE_TOPC:22;
    hence 0*Cl(A) = {}X by CONVEX1:33
      .= Cl(0*A) by A2,A3,CONVEX1:33;
  end;
  suppose
A4: A is non empty;
    then Cl A is non empty by PRE_TOPC:18,XBOOLE_1:3;
    hence 0*Cl(A) = {0.X} by CONVEX1:34
      .= Cl {0.X} by A1,YELLOW_8:26
      .= Cl(0*A) by A4,CONVEX1:34;
  end;
end;
