
theorem Th39:
  for X being non empty TopSpace holds
  CC_0_Functions X is linearly-closed
proof
  let X be non empty TopSpace;
  set Y = CC_0_Functions X;
  set V = ComplexVectSpace(the carrier of X);
A1: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
A2:   v in Y & u in Y;
    reconsider v1=v, u1=u as Element of Funcs((the carrier of X),COMPLEX);
    reconsider v2=v, u2=u as VECTOR of CAlgebra the carrier of X;
    v2+u2 in Y by A2,Lm10;
    hence thesis;
  end;
A3:for a be Complex, v be Element of V st v in Y holds a * v in Y
  proof
    let a be Complex, v be VECTOR of V;
    assume
A4:   v in Y;
    reconsider v1=v as Element of Funcs((the carrier of X),COMPLEX);
    reconsider v2=v as VECTOR of CAlgebra the carrier of X;
    a*v2 in Y by A4,Lm11;
    hence thesis;
  end;
  thus thesis by A1,A3;
end;
