
theorem Th28:
  for X be non empty TopSpace holds
  C_0_Functions(X) is linearly-closed
proof
  let X be non empty TopSpace;
  set Y = C_0_Functions(X);
  set V = RealVectSpace(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),REAL);
    reconsider v2=v, u2=u as VECTOR of RAlgebra the carrier of X;
    v2+u2 in Y by A2,Lm10;
    hence thesis;
  end;
  for a be Real, v be Element of V st v in Y holds a * v in Y
  proof
    let a be Real,v be VECTOR of V;
    assume
A3: v in Y;
    reconsider v1=v as Element of Funcs((the carrier of X),REAL);
    reconsider v2=v as VECTOR of RAlgebra the carrier of X;
    a*v2 in Y by A3,Lm11;
    hence thesis;
  end;
  hence thesis by A1,RLSUB_1:def 1;
end;
