reserve p, q, x, y for Real,
  n for Nat;
reserve X for non empty TopSpace,
  a, b, c, d, e, f for Point of X,
  T for non empty pathwise_connected TopSpace,
  a1, b1, c1, d1, e1, f1 for Point of T;
reserve x0, x1 for Point of X,
  P, Q for Path of x0,x1,
  y0, y1 for Point of T,
  R, V for Path of y0,y1;

theorem Th60:
  for a being Point of TOP-REAL n, C being Loop of a holds the
  carrier of pi_1(TOP-REAL n,a) = { Class(EqRel(TOP-REAL n,a),C) }
proof
  let a be Point of TOP-REAL n, C be Loop of a;
  set X = TOP-REAL n;
  set E = EqRel(X,a);
  hereby
    let x be object;
    assume x in the carrier of pi_1(X,a);
    then consider P being Loop of a such that
A1: x = Class(E,P) by Th47;
    P,C are_homotopic by Th59;
    then x = Class(E,C) by A1,Th46;
    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 Def1;
  then x in Class E by A2,EQREL_1:def 3;
  hence thesis by Def5;
end;
