
theorem Th38:
  for X being non empty TopSpace holds
  CC_0_Functions X is add-closed
proof
  let X be non empty TopSpace;
  set Y = CC_0_Functions X;
  set V = ComplexVectSpace(the carrier of X);
  for v,u be VECTOR of V st v in Y & u in Y holds v + u in Y
  proof
    let v,u be VECTOR of V;
    assume
A1:   v in Y & u in Y;
    reconsider v2=v, u2=u as VECTOR of CAlgebra the carrier of X;
    v2+u2 in Y by A1,Lm10;
    hence thesis;
  end;
hence thesis by IDEAL_1:def 1;
end;
