 reserve
  S for non empty TopSpace,
  T for LinearTopSpace,
  X for non empty Subset of the carrier of S;

theorem Th5:
  for S be non empty TopSpace, T be non empty LinearTopSpace
    holds ContinuousFunctions(S,T) is linearly-closed
proof
  let S be non empty TopSpace, T be non empty LinearTopSpace;
  set W = ContinuousFunctions(S,T);
A1: for v,u be VECTOR of
RealVectSpace(the carrier of S,T) st
  v in ContinuousFunctions(S,T)
& u in ContinuousFunctions(S,T)
holds v + u in ContinuousFunctions  (S,T)
  proof
    let v,u be VECTOR of RealVectSpace(the carrier of S,T) such that
A2: v in W and
A3: u in W;
    consider f be Function of the carrier of S, the carrier of T such that
 A4:v=f & f is continuous by A2;
  consider g be Function of the carrier of S, the carrier of T such that
 A5:u=g & g is continuous by A3;
reconsider u1 = u as Element of Funcs (the carrier of S,the carrier of T);
reconsider v1 = v as Element of Funcs (the carrier of S,the carrier of T);
reconsider h = f+g as Element of
Funcs (the carrier of S,the carrier of T) by FUNCT_2:8;
A6: dom (f+g) = (dom f) /\ (dom g) by VFUNCT_1:def 1
 .= (the carrier of S) /\ (dom g) by FUNCT_2:def 1
 .= (the carrier of S) /\ (the carrier of S) by FUNCT_2:def 1
 .= the carrier of S;
for x be Element of the carrier of S holds h.x = v1.x + u1.x
proof
 let x be Element of the carrier of S;
 thus h.x = h/.x
      .=f/.x + g/.x by VFUNCT_1:def 1,A6
      .=v1.x + u1.x by A4,A5;
end; then
 A7: h = v+u by LOPBAN_1:1;
 f+g is continuous by A4,A5,Th3;
 hence v+u in W by A7;
end;
  for a be Real
  for v be VECTOR of RealVectSpace(the carrier of S,T) st v
  in W holds a * v in W
  proof
    let a be Real;
    let v be VECTOR of RealVectSpace(the carrier of S,T) such that
A8: v in W;
  consider f be Function of the carrier of S, the carrier of T such that
 A9:v=f & f is continuous by A8;
reconsider v1 = v as Element of Funcs (the carrier of S,the carrier of T);
reconsider h = a(#)f as Element of Funcs (the carrier of S,the carrier of T)
  by FUNCT_2:8;
A10: dom (a(#)f) = the carrier of S by FUNCT_2:def 1;
for x be Element of the carrier of S holds h.x = a*v1.x
proof
 let x be Element of the carrier of S;
 thus h.x = h/.x
      .=a*f/.x by VFUNCT_1:def 4,A10
      .=a*v1.x by A9;
end; then
 A11: h = a*v by LOPBAN_1:2;
 a(#)f is continuous by A9,Th4;
 hence a*v in W by A11;
end;
hence thesis by A1;
end;
