reserve T,U for non empty TopSpace;
reserve t for Point of T;
reserve n for Nat;
reserve T for TopStruct;
reserve f for PartFunc of R^1, T;
reserve c for Curve of T;
reserve T for non empty TopStruct;

theorem
  for n being non zero Nat, r being positive Real,
      x being Point of TOP-REAL n st n >= 3 holds
  Tcircle(x,r) is having_trivial_Fundamental_Group
  proof
    let n be non zero Nat;
    let r be positive Real;
    let x be Point of TOP-REAL n;
    assume
A1: n >= 3;
    then n-1 >= 3-1 by XREAL_1:9;
    then 0 <= n-1 by XXREAL_0:2;
    then
A2: n-'1+1 = n-1+1 by XREAL_0:def 2;
    2+1 = 3;
    then 2 <= n-'1 by A1,NAT_D:49;
    then
A3: TUnitSphere(n-'1) is having_trivial_Fundamental_Group by Th46;
A4: TUnitSphere(n-'1) = Tunit_circle(n-'1+1) by MFOLD_2:def 4;
A5: Tunit_circle(n) = Tcircle(0.TOP-REAL n,1) by TOPREALB:def 7;
    n in NAT by ORDINAL1:def 12;
    then Tcircle(x,r), Tcircle(0.TOP-REAL n,1) are_homeomorphic by TOPREALB:20;
    hence thesis by A2,A3,A4,A5,Th13;
  end;
