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

theorem Th2:
for X be non empty TopSpace, S be non empty LinearTopSpace,
    f be Function of X,S,
    x be Point of X,
    a be Real
  st f is_continuous_at x
holds a(#)f is_continuous_at x
proof
let X be non empty TopSpace, S be non empty LinearTopSpace,
     f be Function of X,S,
     x be Point of X,
     a be Real;
assume A1: f is_continuous_at x;
for G being a_neighborhood of (a(#)f) . x
ex H being a_neighborhood of x st
(a(#)f) .: H c= G
proof
  let G be a_neighborhood of (a(#)f) . x;
A2: dom (a(#)f) = the carrier of X by FUNCT_2:def 1;
A3: ((a(#)f)) . x =((a(#)f))/. x
          .=a*f/.x by VFUNCT_1:def 4,A2
          .=a*f.x;
consider W being Subset of S such that
A4: W is open & W c= G & ((a(#)f)).x in W by CONNSP_2:6;
consider r be positive Real,W1 being Subset of S such that
A5:W1 is open & f.x in W1
& for s being Real st |.(s - a).| < r holds
   s * W1 c= W by A3,A4,RLTOPSP1:def 9;
 |.a-a.| = 0 by COMPLEX1:61; then
A6: a * W1 c= W by A5;
reconsider W1 as a_neighborhood of f.x by A5,CONNSP_2:3;
consider H1 being a_neighborhood of x such that
A7: f .: H1 c= W1 by TMAP_1:def 2,A1;
take H1;
thus (a(#)f) .: H1 c= G
proof
  let y be object;
  assume y in (a(#)f ) .: H1; then
  consider z be object such that
  A8: z in the carrier of X
    & z in H1
    & y=(a(#)f).z by FUNCT_2:64;
  reconsider z as Point of X by A8;
A9: (a(#)f) . z =(a(#)f)/. z
          .=a*f/.z by VFUNCT_1:def 4,A2
          .=a*f.z;
  f.z in f.:H1 by FUNCT_2:35,A8; then
  a*f.z in a*W1 by A7;
  hence y in G by A4,A6,A8,A9;
end;
end;
hence thesis by TMAP_1:def 2;
end;
