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 Th62:
  for C being constant Loop of a holds 1_pi_1(X,a) = Class(EqRel(X ,a),C)
proof
  let C be constant Loop of a;
  set G = pi_1(X,a);
  reconsider g = Class(EqRel(X,a),C) as Element of G by Th47;
  set E = EqRel(X,a);
  now
    let h be Element of G;
    consider P being Loop of a such that
A1: h = Class(E,P) by Th47;
A2: P,P+C are_homotopic by BORSUK_6:80;
    thus h * g = Class(E,P+C) by A1,Lm4
      .= h by A1,A2,Th46;
A3: P,C+P are_homotopic by BORSUK_6:82;
    thus g * h = Class(E,C+P) by A1,Lm4
      .= h by A1,A3,Th46;
  end;
  hence thesis by GROUP_1:4;
end;
