reserve a,b,c,h for Integer;
reserve k,m,n for Nat;
reserve i,j,z for Integer;
reserve p for Prime;

theorem Th108:
  for m being Integer
  for x,y,z being non zero Integer st
  x/y + y/z + z/x = m & x,y,z are_mutually_coprime holds
  (x = 1 or x = -1) & (y = 1 or y = -1) & (z = 1 or z = -1)
  proof
    let m be Integer;
    let x,y,z be non zero Integer;
    assume x/y + y/z + z/x = m;
    then
A1: x^2*z + y^2*x + z^2*y = m*x*y*z by Th107;
    assume
A2: x,y,z are_mutually_coprime;
A3: y divides m*x*z*y;
A4: y divides z^2*y;
    y divides x*y*y;
    then y divides m*x*z*y - y^2*x by A3,INT_5:1;
    then
A5: y divides m*x*z*y - y^2*x - z^2*y by A4,INT_5:1;
A6: z divides m*x*y*z;
A7: z divides y*z*z;
    z divides x^2*z;
    then z divides m*x*y*z - x^2*z by A6,INT_5:1;
    then
A8: z divides m*x*y*z - x^2*z - z^2*y by A7,INT_5:1;
A9: x divides m*y*z*x;
A10: x divides x*z*x;
    x divides y^2*x;
    then x divides m*y*z*x - y^2*x by A9,INT_5:1;
    then
A11: x divides m*y*z*x - y^2*x - x^2*z by A10,INT_5:1;
    z^2 = z|^2 by WSIERP_1:1;
    then z^2,x are_coprime by A2,WSIERP_1:10;
    hence x = 1 or x = -1 by A1,A2,A11,INT_2:25,Th14;
    x^2 = x|^2 by WSIERP_1:1;
    then x^2,y are_coprime by A2,WSIERP_1:10;
    hence y = 1 or y = -1 by A1,A2,A5,INT_2:25,Th14;
    y^2 = y|^2 by WSIERP_1:1;
    then y^2,z are_coprime by A2,WSIERP_1:10;
    hence z = 1 or z = -1 by A1,A2,A8,INT_2:25,Th14;
  end;
