reserve a,b for Complex;
reserve V,X,Y for ComplexLinearSpace;
reserve u,u1,u2,v,v1,v2 for VECTOR of V;
reserve z,z1,z2 for Complex;
reserve V1,V2,V3 for Subset of V;

theorem Th23:
  {0.V} is linearly-closed
proof
  thus for v,u being VECTOR of V st v in {0.V} & u in {0.V} holds v + u in {0.
  V}
  proof
    let v,u be VECTOR of V;
    assume v in {0.V} & u in {0.V};
    then v = 0.V & u = 0.V by TARSKI:def 1;
    then v + u = 0.V by RLVECT_1:4;
    hence thesis by TARSKI:def 1;
  end;
  let z;
  let v be VECTOR of V;
  assume
A1: v in {0.V};
  then v = 0.V by TARSKI:def 1;
  hence thesis by A1,Th1;
end;
