
theorem Th18:
  for T being non empty TopStruct, A being Subset of T holds A c= Cl_Seq(A)
proof
  let T be non empty TopStruct, A be Subset of T;
  let x be object;
  assume
A1: x in A;
  then reconsider x9=x as Point of T;
  ex S being sequence of T st rng S c= A & x9 in Lim S
  proof
    set S = NAT --> x9;
    take S;
    {x9} c= A
    by A1,TARSKI:def 1;
    hence rng S c= A by FUNCOP_1:8;
    S is_convergent_to x9 by FRECHET:22;
    hence thesis by FRECHET:def 5;
  end;
  hence thesis by Def1;
end;
