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 Th53:
  x0,x1 are_connected implies (pi_1-iso(P))" = pi_1-iso(-P)
proof
  set f = pi_1-iso(P);
  set g = pi_1-iso(-P);
  assume
A1: x0,x1 are_connected;
    then f is one-to-one onto by Th51,Th52;
    then
A2: f" = (f qua Function)" by TOPS_2:def 4;
A3: f is one-to-one by A1,Th51;
  for x being Element of pi_1(X,x0) holds f".x = g.x
  proof
    let x be Element of pi_1(X,x0);
    consider Q being Loop of x0 such that
A4: x = Class(EqRel(X,x0),Q) by Th47;
    --P = P by A1,BORSUK_6:43;
    then
A5: P+(-P+Q+--P)+-P, Q are_homotopic by A1,Th41;
    dom f = the carrier of pi_1(X,x1) by FUNCT_2:def 1;
    then
A6: Class(EqRel(X,x1),-P+Q+--P) in dom f by Th47;
    f.Class(EqRel(X,x1),-P+Q+--P) = Class(EqRel(X,x0),P+(-P+Q+--P)+-P) by A1
,Def6
      .= x by A4,A5,Th46;
    hence f".x = Class(EqRel(X,x1),-P+Q+--P) by A3,A2,A6,FUNCT_1:32
      .= g.x by A1,A4,Def6;
  end;
  hence thesis;
end;
