reserve n for Element of NAT,
  a, b for Real;

theorem Th13:
  for T being non empty interval SubSpace of R^1, a being Point of T
  , C being Loop of a holds the carrier of pi_1(T,a) = { Class(EqRel(T,a),C) }
proof
  let T be non empty interval SubSpace of R^1, a be Point of T, C be Loop of a;
  set E = EqRel(T,a);
  hereby
    let x be object;
    assume x in the carrier of pi_1(T,a);
    then consider P being Loop of a such that
A1: x = Class(E,P) by TOPALG_1:47;
    P,C are_homotopic by Th12;
    then x = Class(E,C) by A1,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
A2: x = Class(E,C) by TARSKI:def 1;
  C in Loops a by TOPALG_1:def 1;
  then x in Class E by A2,EQREL_1:def 3;
  hence thesis by TOPALG_1:def 5;
end;
