reserve T,U for non empty TopSpace;
reserve t for Point of T;
reserve n for Nat;

theorem Th17:
  (for t being Point of T, f being Loop of t holds f is nullhomotopic)
  implies T is having_trivial_Fundamental_Group
  proof
    assume
A1: for t being Point of T, f being Loop of t holds f is nullhomotopic;
    for t being Point of T holds pi_1(T,t) is trivial
    proof
      let t be Point of T;
      set C = the constant Loop of t;
      the carrier of pi_1(T,t) = { Class(EqRel(T,t),C) }
      proof
        set E = EqRel(T,t);
        hereby
          let x be object;
          assume x in the carrier of pi_1(T,t);
          then consider P be Loop of t such that
    A2:   x = Class(E,P) by TOPALG_1:47;
          P is nullhomotopic by A1;
          then consider C1 be constant Loop of t such that
    A3:   P, C1 are_homotopic;
          C1, C are_homotopic by Th5;
          then P, C are_homotopic by A3,BORSUK_6:79;
          then x = Class(E,C) by A2,TOPALG_1:46;
          hence x in { Class(E,C) } by TARSKI:def 1;
        end;
        let x be object;
        assume x in { Class(E,C) };
        then
    A4: x = Class(E,C) by TARSKI:def 1;
        C in Loops t by TOPALG_1:def 1;
        then x in Class E by A4,EQREL_1:def 3;
        hence thesis by TOPALG_1:def 5;
      end;
      hence thesis;
    end;
    hence thesis;
  end;
