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

theorem Th15:
  for T being having_trivial_Fundamental_Group non empty TopSpace,
      t being Point of T, P1,P2 being Loop of t
  holds P1,P2 are_homotopic
  proof
    let T be having_trivial_Fundamental_Group non empty TopSpace;
    let t be Point of T, P1,P2 be Loop of t;
    Class(EqRel(T,t),P1) in the carrier of pi_1(T,t) &
    Class(EqRel(T,t),P2) in the carrier of pi_1(T,t) by TOPALG_1:47;
    then Class(EqRel(T,t),P1) = Class(EqRel(T,t),P2) by ZFMISC_1:def 10;
    hence thesis by TOPALG_1:46;
  end;
