
theorem Th20:
  for T being non empty TopStruct holds T is Frechet iff for A
  being Subset of T holds Cl(A)=Cl_Seq(A)
proof
  let T be non empty TopStruct;
  thus T is Frechet implies for A being Subset of T holds Cl(A)=Cl_Seq(A)
  proof
    assume
A1: T is Frechet;
    let A be Subset of T;
    reconsider A9=A as Subset of T;
    thus Cl(A)c=Cl_Seq(A)
    proof
      let x be object;
      assume
A2:   x in Cl(A);
      then reconsider x9=x as Point of T;
      ex S being sequence of T st rng S c= A9 & x9 in Lim S by A1,A2;
      hence thesis by Def1;
    end;
    thus thesis by Th14;
  end;
  assume
A3: for A being Subset of T holds Cl(A)=Cl_Seq(A);
  let A be Subset of T,x be Point of T;
  assume x in Cl(A);
  then x in Cl_Seq(A) by A3;
  hence thesis by Def1;
end;
