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 Th51:
  x0,x1 are_connected implies pi_1-iso(P) is one-to-one
proof
  assume
A1: x0,x1 are_connected;
  set f = pi_1-iso(P);
  let a, b be object such that
A2: a in dom f and
A3: b in dom f and
A4: f.a = f.b;
  consider B being Loop of x1 such that
A5: b = Class(EqRel(X,x1),B) by A3,Th47;
A6: f.b = Class(EqRel(X,x0),P+B+-P) by A1,A5,Def6;
  consider A being Loop of x1 such that
A7: a = Class(EqRel(X,x1),A) by A2,Th47;
  f.a = Class(EqRel(X,x0),P+A+-P) by A1,A7,Def6;
  then P+A+-P, P+B+-P are_homotopic by A4,A6,Th46;
  then P+A, P+B are_homotopic by A1,Th27;
  then A,B are_homotopic by A1,Th29;
  hence thesis by A7,A5,Th46;
end;
