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
  for x, y being Element of pi_1(X,a), P being Loop of a st x = Class(
  EqRel(X,a),P) & y = Class(EqRel(X,a),-P) holds x" = y
proof
  set E = EqRel(X,a);
  set G = pi_1(X,a);
  let x, y be Element of G, P be Loop of a such that
A1: x = Class(E,P) & y = Class(E,-P);
  set C = the constant Loop of a;
A2: -P+P,C are_homotopic by BORSUK_6:86;
A3: y * x = Class(E,-P+P) by A1,Lm4
    .= Class(E,C) by A2,Th46
    .= 1_G by Th62;
A4: P+-P,C are_homotopic by BORSUK_6:84;
  x * y = Class(E,P+-P) by A1,Lm4
    .= Class(E,C) by A4,Th46
    .= 1_G by Th62;
  hence thesis by A3,GROUP_1:def 5;
end;
