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 Th48:
  x0,x1 are_connected & P,Q are_homotopic implies pi_1-iso(P) = pi_1-iso(Q)
proof
  assume that
A1: x0,x1 are_connected and
A2: P,Q are_homotopic;
  for x being Element of pi_1(X,x1) holds (pi_1-iso(P)).x = (pi_1-iso(Q)). x
  proof
A3: -P, -Q are_homotopic by A1,A2,BORSUK_6:77;
    let x be Element of pi_1(X,x1);
    consider L being Loop of x1 such that
A4: x = Class(EqRel(X,x1),L) by Th47;
    L, L are_homotopic by BORSUK_2:12;
    then P+L, Q+L are_homotopic by A1,A2,BORSUK_6:75;
    then
A5: P+L+-P, Q+L+-Q are_homotopic by A1,A3,BORSUK_6:75;
    thus (pi_1-iso(P)).x = Class(EqRel(X,x0),P+L+-P) by A1,A4,Def6
      .= Class(EqRel(X,x0),Q+L+-Q) by A5,Th46
      .= (pi_1-iso(Q)).x by A1,A4,Def6;
  end;
  hence thesis;
end;
