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

theorem Th1:
for X be non empty TopSpace, S be non empty LinearTopSpace,
    f,g be Function of X,S,
    x be Point of X
  st f is_continuous_at x & g is_continuous_at x
holds f+g is_continuous_at x
proof
let X be non empty TopSpace, S be non empty LinearTopSpace,
     f,g be Function of X,S,
     x be Point of X;
assume A1: f is_continuous_at x & g is_continuous_at x;
for G being a_neighborhood of (f+g) . x
ex H being a_neighborhood of x st
(f+g) .: H c= G
proof
let G be a_neighborhood of (f+g) . x;
A2: dom f = the carrier of X by FUNCT_2:def 1;
A3: dom g = the carrier of X by FUNCT_2:def 1;
A4: dom (f+g) = (dom f) /\ (dom g) by VFUNCT_1:def 1
             .= the carrier of X by A2,A3;
A5:(f+g) . x =(f+g)/. x
          .=f/.x+g/.x by VFUNCT_1:def 1,A4
          .=f.x +g.x;
consider W being Subset of S such that
A6: W is open & W c= G & (f+g).x in W by CONNSP_2:6;
consider W1, W2 being Subset of S such that
A7: W1 is open & W2 is open & f.x in W1 & g.x in W2
  & W1 + W2 c= W by A5,A6,RLTOPSP1:def 8;
reconsider W1 as a_neighborhood of f.x by A7,CONNSP_2:3;
reconsider W2 as a_neighborhood of g.x by A7,CONNSP_2:3;
consider H1 being a_neighborhood of x such that
A8: f .: H1 c= W1 by TMAP_1:def 2,A1;
consider H2 being a_neighborhood of x such that
A9: g .: H2 c= W2 by TMAP_1:def 2,A1;
reconsider H = H1/\ H2 as a_neighborhood of x by CONNSP_2:2;
take H;
thus (f+g) .: H c= G
proof
  let y be object;
  assume y in (f+g) .: H; then
  consider z be object such that
  A10: z in the carrier of X
    & z in H
    & y=(f+g).z by FUNCT_2:64;
 reconsider z as Point of X by A10;
A11: (f+g) . z =(f+g)/. z
          .=f/.z+g/.z by VFUNCT_1:def 1,A4
          .=f.z +g.z;
  z in H1 by XBOOLE_0:def 4,A10; then
A12: f.z in f.:H1 by FUNCT_2:35;
  z in H2 by XBOOLE_0:def 4,A10; then
  g.z in g.:H2 by FUNCT_2:35; then
  f.z+g.z in W1+W2 by A12,A8,A9;
  hence y in G by A6,A7,A10,A11;
end;
end;
hence thesis by TMAP_1:def 2;
end;
