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

theorem Th12:
  T is simply_connected iff
  for t1,t2 being Point of T holds t1,t2 are_connected &
  for P, Q being Path of t1,t2 holds
  Class(EqRel(T,t1,t2),P) = Class(EqRel(T,t1,t2),Q)
  proof
    hereby
      assume
A1:   T is simply_connected;
      let t1,t2 be Point of T;
      thus
A2:   t1,t2 are_connected by A1,BORSUK_2:def 3;
      let P, Q be Path of t1,t2;
      set E = EqRel(T,t1,t2);
A3:   P,P+-Q+Q are_homotopic by A1,TOPALG_1:22;
      set C = the constant Loop of t1;
      P+-Q,C are_homotopic by A1,Th11;
      then
A4:   P+-Q+Q,C+Q are_homotopic by A1,BORSUK_6:76;
      C+Q,Q are_homotopic by A1,BORSUK_6:83;
      then P+-Q+Q,Q are_homotopic by A4,BORSUK_6:79;
      then P,Q are_homotopic by A3,BORSUK_6:79;
      hence Class(E,P) = Class(E,Q) by A2,TOPALG_1:46;
    end;
    assume
A5: for t1,t2 being Point of T holds t1,t2 are_connected &
    for P, Q being Path of t1,t2 holds
    Class(EqRel(T,t1,t2),P) = Class(EqRel(T,t1,t2),Q);
    thus T is having_trivial_Fundamental_Group
    proof
      let t be Point of T;
      let x, y be Element of pi_1(T,t);
      (ex P being Loop of t st x = Class(EqRel(T,t),P)) &
      ex P being Loop of t st y = Class(EqRel(T,t),P) by TOPALG_1:47;
      hence thesis by A5;
    end;
    thus thesis by A5;
end;
