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

theorem
  { [x,y,z] where x,y,z is non zero Integer:
     x/y + y/z + z/x = 3 & x,y,z are_mutually_coprime } =
  { [1,1,1] , [-1,-1,-1] }
  proof
    set A = { [x,y,z] where x,y,z is non zero Integer:
    x/y + y/z + z/x = 3 & x,y,z are_mutually_coprime };
    set B = { [1,1,1] , [-1,-1,-1] };
    thus A c= B
    proof
      let a be object;
      assume a in A;
      then consider x,y,z being non zero Integer such that
A1:   a = [x,y,z] and
A2:   x/y + y/z + z/x = 3 and
A3:   x,y,z are_mutually_coprime;
      (x = 1 or x = -1) & (y = 1 or y = -1) & (z = 1 or z = -1)
      by A2,A3,Th108;
      hence thesis by A1,A2,TARSKI:def 2;
    end;
    let a be object;
    assume a in B;
    then
A4: a = [1,1,1] or a = [-1,-1,-1] by TARSKI:def 2;
A5: 1,1,1 are_mutually_coprime by WSIERP_1:9;
    -1,-1,-1 are_mutually_coprime by Th11,WSIERP_1:9;
    hence thesis by A4,A5,Lm14,Lm15;
   end;
