reserve r,s,t,u for Real;

theorem Th31:
  for X being LinearTopSpace, x1,x2 being Point of X, V being
  a_neighborhood of x1+x2 ex V1 being a_neighborhood of x1, V2 being
  a_neighborhood of x2 st V1+V2 c= V
proof
  let X be LinearTopSpace;
  let x1,x2 be Point of X, V be a_neighborhood of x1+x2;
  x1+x2 in Int V by CONNSP_2:def 1;
  then consider V1,V2 being Subset of X such that
A1: V1 is open and
A2: V2 is open and
A3: x1 in V1 and
A4: x2 in V2 and
A5: V1+V2 c= Int V by Def8;
  Int V2 = V2 by A2,TOPS_1:23;
  then reconsider V2 as a_neighborhood of x2 by A4,CONNSP_2:def 1;
  Int V1 = V1 by A1,TOPS_1:23;
  then reconsider V1 as a_neighborhood of x1 by A3,CONNSP_2:def 1;
  take V1,V2;
  Int V c= V by TOPS_1:16;
  hence thesis by A5;
end;
